arXiv:1402.3482v4 [math.QA] 11 Feb 2015
ON UNIMODULAR FINITE TENSOR CATEGORIES
KENICHI SHIMIZU
Abstract. Let C be a finite tensor category with simple unit object, let Z(C)
denote its monoidal center, and let L and R be a left adjoint and a right
adjoint of the forgetful functor U : Z(C) → C. We show that the following
conditions are equivalent: (1) C is unimodular, (2) U is a Frobenius functor, (3)
L preserves the duality, (4) R preserves the duality, (5) L(1) is self-dual, and
(6) R(1) is self-dual, where 1 ∈ C is the unit object. We also give some other
equivalent conditions. As an application, we give a categorical understanding
of some topological invariants arising from finite-dimensional unimodular Hopf
algebras.
1. Introduction
For a locally compact group G with right Haar measure µ, the modular function
is defined as the unique function α : G → R+ such that µ(gE) = α(g)µ(E) for
all Borel subsets E of G. If the modular function is constantly one, then G is
said to be unimodular. We can define the modular function (usually called the
distinguished grouplike element) and the unimodularity of a Hopf algebra by using
the intregral theory for Hopf algebras instead of Haar measures. They are important
not only in the Hopf algebra theory (e.g., the Radford S 4 -formula [32]), but also in
their applications to topology: For example, given a finite-dimensional unimodular
ribbon Hopf algebra, one can construct an invariant of closed 3-manifolds [21, 25].
Recently, Ishii and Masuoka [23] developed a method to construct an invariant of
handlebody-links from finite-dimensional unimodular Hopf algebras.
A finite tensor category [19] is a class of monoidal categories including the representation category of a finite-dimensional Hopf algebra. To generalize the Radford
S 4 -formula to finite tensor categories, Etingof, Nikshych and Ostrik [17] introduced
the distinguished invertible object D ∈ C. This object is a categorical analogue of
the modular function, and therefore C is said to be unimodular if D ∼
= 1. In this
paper, we investigate the object D in detail and provide some characterizations of
the unimodularity of a finite tensor category by using the monoidal center. As an
application, we give a categorical understanding of the above-mentioned constructions of topological invariants by generalizing such constructions to unimodular
finite tensor categories.
This paper is organized as follows: In Section 2, we recall from [16, 19, 28, 24]
some basic results on finite tensor categories and their module categories. We warn
that, unlike [19], we do not assume that the unit object of a finite tensor category is
simple (see §2.7 for our definition). In relation to this, we note that some additional
technical assumptions on finite tensor categories will be made at the beginning of
each of Sections 4, 5 and 6.
In Section 3, we first recall from [28] the notions of ends and coends. Following
[6, 10], the (monoidal) center Z(C) of a rigid monoidal category C is isomorphic to
1
2
KENICHI SHIMIZU
the category of modules over a certain Hopf monad on C, which we call the central
Hopf monad on C, provided that the coend
Z X∈C
X∗ ⊗ V ⊗ X
(1.1)
Z(V ) =
exists for all V ∈ C. We also show that a coend of certain type of functors, including (1.1), exists in a finite tensor category. As an application, we give an alternative
proof of the fact that the center of a finite tensor category is again a finite tensor
category [19].
Our main theorem is proved in Section 4. There is an algebra A ∈ C ⊠ C rev which
plays a crucial role in the definition of the distinguished invertible object of a finite
tensor category C. We express the algebra A as a coend of a certain functor, and
then show its relation with the central Hopf monad on C. Based on this observation,
e such that the diagram
we see that there are equivalences K and K
Z(C)
e
K
/ (the category of A-bimodules in C ⊠ C rev )
FA
U
C
K
/ (the category of right A-modules in C ⊠ C rev )
commutes, where FA is the functor forgetting the left A-module structure. Now let
L and R be a left and a right adjoint functor of U . By using the above commutative
diagram, we obtain a natural isomorphism
(1.2)
R(V ) ∼
= L(D ⊗ V ) (V ∈ C),
where D ∈ C is the distinguished invertible object of C (Lemma 4.7). Once (1.2) is
obtained, the following our main theorem is proved without much difficulty:
Theorem (Theorem 4.10). The following assertions are equivalent:
(1) C is unimodular.
(2) U is a Frobenius functor, i.e., L ∼
= R.
(3) L preserves the left duality, i.e., there exists a natural isomorphism L(V ∗ ) ∼
=
L(V )∗ for V ∈ C, where (−)∗ is the left duality on C.
(4) R preserves the left duality.
If, moreover, the unit object 1 ∈ C is a simple object, then the above conditions are
equivalent to each of the following conditions:
(5) L(1) ∼
= L(1)∗ .
(6) R(1) ∼
= R(1)∗ .
(7) HomZ(C) (1, L(1)) 6= 0.
(8) HomZ(C) (R(1), 1) 6= 0.
We note that the equivalence between (1) and (2) has been obtained by Caenepeel,
Militaru and Zhu in [7, §4, Theorem 53] in the case where C is the category of representations of a finite-dimensional Hopf algebra.
In Section 5, we apply our techniques to investigate further properties of the
distinguished invertible object. Here we give a new proof of the Radford S 4 -formula
(§5.2), determine when L and R are faithful (Theorem 5.2), and introduce a formula
expressing D as an end of a certain functor (Lemma 5.4). As an application of the
formula of D, we show that every semisimple finite tensor category (= a multi-fusion
category [18]) is unimodular, as conjectured in [12].
ON UNIMODULAR FINITE TENSOR CATEGORIES
3
In Section 6, we apply our results to study the role of the unimodularity in
the constructions of some topological invariants (for this reason, in this section, we
always assume EndC (1) ∼
= k). It is known that B := R(1) is a commutative algebra
in Z(C). The unimodularity can be characterized by this algebra:
Theorem (Theorem 6.1). B is Frobenius if and only if C is unimodular.
As a first application of this theorem, in §6.3, we give a categorical understanding
of Ishii and Masuoka’s construction of handlebody-link invariants [23] by generalizing their construction in the setting of unimodular finite tensor categories. The
second application concerns the object Int(FC ) of integrals of a certain Hopf algebra FC in a braided finite tensor category C, which is used to construct 3-manifold
invariants in [27, 36]. We prove:
Theorem (Theorem 6.8). Int(FC ) ∼
= D∗ .
Finally, we suppose that C is a unimodular ribbon finite tensor category. Then
the above theorem implies that a non-zero integral Λ of FC is an algebraic Kirby
element (in the sense of Virelizier [36]). Hence Λ gives rise to a closed 3-manifold
invariant that generalizes the Hennings-Kauffman-Radford invariant constructed
from a finite-dimensional unimodular ribbon Hopf algebra (Remark 6.10).
Acknowledgments
A part of this work is done during a visit of the author to Université de Bourgogne
in November 2013. The author is grateful to Peter Schauenburg for his hospitality
and for helpful discussion. The author also would like to thank Atsushi Ishii, Akira
Masuoka, and Taiki Shibata for helpful discussion. The author is supported by
Grant-in-Aid for JSPS Fellows (24·3606).
2. Preliminaries
2.1. Monoidal categories. For the basic theory of monoidal categories, we refer
the reader to [2, 24, 28]. We first fix some conventions used throughout this paper.
In view of Mac Lane’s coherence theorem, we may, and do, assume that all monoidal
categories are strict. Given a monoidal category C = (C, ⊗, 1) with tensor product
⊗ and unit object 1 ∈ C, we set
C op = (C op , ⊗, 1) and
C rev = (C, ⊗rev , 1),
where Mop for a category M means the opposite category and ⊗rev is the reversed
tensor product given by V ⊗rev W = W ⊗ V for V, W ∈ C.
Let C and D be monoidal categories. A monoidal functor from C to D is a functor
F : C → D endowed with a morphism F0 : 1 → F (1) and a natural transformation
F2 (V, W ) : F (V ) ⊗ F (W ) → F (V ⊗ W )
(V, W ∈ C)
satisfying certain axioms [28, XI.2]. If F0 and F2 are invertible, then F is said to
be strong. A comonoidal functor is a monoidal functor from C op to Dop .
Following [24], a left dual object of V ∈ C is an object V ∗ ∈ C endowed with
morphisms evV : V ∗ ⊗ V → 1 and coevV : 1 → V ⊗ V ∗ in C such that
(coevV ⊗ idV )(idV ⊗ evV ) = idV
∗
and (evV ⊗ idV ∗ )(idV ∗ ⊗ coevV ) = idV ∗ .
One can extend V 7→ V to a strong monoidal functor (−)∗ : C op → C rev , called the
left duality functor, provided that every object of C has a left dual object. A right
4
KENICHI SHIMIZU
dual object ∗ V of V ∈ C is a left dual object of V in C rev . Similarly to the above,
one can extend V 7→ ∗ V to a strong monoidal functor ∗ (−) : C op → C rev if every
object of C has a right dual object.
A monoidal category C is said to be rigid (or autonomous) if every object of C has
both a left and a right dual object. If this is the case, the contravariant endofunctors
(−)∗ and ∗ (−) on C are mutually quasi-inverse. Moreover, by replacing C with an
equivalent one, we can choose dual objects so that
1∗ = 1, (V ⊗ W )∗ = W ∗ ⊗ V ∗ and
∗
(V ∗ ) = V = (∗ V )∗
hold for all V, W ∈ C [33]. Thus, throughout this paper, we always assume that
these equations hold.
2.2. Monoidal center. Let C be a monoidal category. A half-braiding for V ∈ C
is a natural isomorphism σV : V ⊗ (−) → (−) ⊗ V such that
σV (X ⊗ Y ) = (idX ⊗ σV (Y )) ◦ (σV (X) ⊗ idY )
holds for all X, Y ∈ C. The monoidal center (or the center for short) of C is the
category Z(C) whose objects are the pairs (V, σV ), where V ∈ C and σV is a halfbraiding for V , and whose morphisms are the morphisms in C compatible with the
half-braidings. The category Z(C) has a natural structure of a braided monoidal
category; see, e.g., [24, XIII.4].
2.3. Algebras in a monoidal category. An algebra (= a monoid [28]) in a
monoidal category C is an object of C endowed with morphisms mA : A ⊗ A → A
and uA : 1 → A obeying the associative law and the unit law. The morphisms mA
and uA are called the multiplication and the unit of A, respectively.
Given an algebra A in C, we denote by A C and CA the categories of left A-modules
and right A-modules, respectively. If M is a left A-module whose underlying object
is left rigid, then its left dual object M ∗ is a right A-module with action
ρ∗
id∗ ⊗evA
M
M ∗ ⊗ A −−−−
−−→ (A ⊗ M )∗ ⊗ A = M ∗ ⊗ A∗ ⊗ A −−−−M
−−−−−−→ M ∗ ,
where ρ : A ⊗ M → M is the left action of A on M . Similarly, a right dual object
of a right A-module has a structure of a left A-module.
Now let B be another algebra in C. If X ∈ A C and Y ∈ CB , then their tensor
product X ⊗ Y is an A-B-bimodule. This construction gives rise to a bifunctor
AC
× CB → A CB ,
(X, Y ) 7→ X ⊗ Y
(X ∈ A C, Y ∈ CB ),
where A CB denotes the category of A-B-bimodules. For simplicity, we now suppose
that C is rigid. The following lemma is well-known:
Lemma 2.1. Let FA : A CB → CB and FB : A CB → A C be the functors forgetting
the actions of A and B, respectively. Then:
(1) A A ⊗ (−) is left adjoint to FA
(2) ∗ (AA ) ⊗ (−) is right adjoint to FA .
(3) (−) ⊗ BB is left adjoint to FB
(4) (−) ⊗ (B B)∗ is right adjoint to FB .
Here, given an algebra A in C, we denote by A A and AA the object A viewed as a
left A-module and a right A-module by the multiplication of A, respectively.
ON UNIMODULAR FINITE TENSOR CATEGORIES
5
For an object K and an algebra A in C, there is a bijection
∼
=
∼
=
Θ : HomA (AA , K ⊗ (A A)∗ ) −−−−−−−−→ HomC (1, K ⊗ A∗ ) −−−−−→ HomC (A, K)
Lemma 2.1
A morphism λ : A → K is called a K-valued trace if Θ−1 (λ) is an isomorphism of
right A-modules. Now we suppose that A has a K-valued trace λ.
Lemma 2.2. Let A, K and λ be as above, and set φ = (Θ−1 (λ))−1 . Then
φ−1
idK ⊗φ∗
ν : A −−−−−−→ K ⊗ A∗ −−−−−−−−−→ K ⊗ A∗∗ ⊗ K ∗
is an isomorphism of algebras in C.
Proof. We denote the multiplication of A and A′ := K ⊗ A∗∗ ⊗ K ∗ by m and m′ ,
respectively. By definition, m′ is given by
∗∗
∗
m′ = (idK ⊗ m∗∗ ⊗ id∗K ) ◦ (idK ⊗ id∗∗
A ⊗ evK ⊗ idA ⊗ idK ).
Since φ is an isomorphism of right A-modules, we have
(φ ⊗ evA ) ◦ (idK ⊗ m∗ ⊗ idA ) = m ◦ (φ ⊗ idA ).
Translating (2.1) via HomC (K ⊗ A∗ ⊗ A, A) ∼
= HomC (K ⊗ A∗ , A ⊗ A∗ ), we get
(2.1)
(2.2)
(φ ⊗ id∗A ) ◦ (idK ⊗ m∗ ) = (m ⊗ id∗A ) ◦ (φ ⊗ coevA ).
Applying the left duality functor to (2.2), we get
(2.3)
∗∗
∗∗
∗
∗
(m∗∗ ⊗ id∗K ) ◦ (id∗∗
A ⊗ φ ) = (evA∗ ⊗ φ ) ◦ (idA ⊗ m ).
One can verify m′ ◦ (ν ⊗ ν) = ν ◦ m directly by using (2.1)–(2.3). Figure 1 explains
the details of the computation graphically (we read string diagrams from the top
to the bottom and express the evaluation and the coevaluation by a cup and a cap,
respectively). It is obvious that ν is invertible. Thus that ν preserves the unit
follows from the uniqueness of the unit.
A 1-valued trace is simply called a trace. Recall that a functor F is said to be
Frobenius [7] if it has a left adjoint functor which is also right adjoint to F . The
following result is an immediate consequence of Lemma 2.1.
Lemma 2.3. For an algebra A in C, the following assertions are equivalent:
(1) A trace for A exists.
(2) AA ∼
= (A A)∗ as right A-modules.
(3) The forgetful functor CA → C is a Frobenius functor.
A Frobenius algebra in C is a pair (A, λ) consisting of an algebra A in C and
a trace λ : A → 1. By abuse of terminology, we often say that an algebra A is
Frobenius if the equivalent conditions of Lemma 2.3 are satisfied.
2.4. Colax-lax adjunctions. The category Set of all sets is a monoidal category
with respect to the Cartesian product and with unit object the set {∗} consisting
of one element. Now let A, B and C be monoidal categories. If P : A → C is a
comonoidal functor and Q : B → C is a monoidal functor, then
H : Aop × B → Set,
(V, W ) 7→ HomC (P (V ), Q(W ))
(V ∈ A, W ∈ B)
has a structure of a monoidal functor given by H0 (∗) = Q0 ◦ P0 and
H2 : H(V, W ) × H(X, Y ) → H(V ⊗ X, W ⊗ Y ),
(f, g) 7→ Q2 (W, Y ) ◦ (f ⊗ g) ◦ P2 (V, X).
6
KENICHI SHIMIZU
(2.3)
=
=
(2.2)
=
=
(2.1)
=
Figure 1. The proof of Lemma 2.2
Following Mac Lane [28, IV], we write
(2.4)
hF, G, η, εi : B ⇀ C
if F : B → C is a functor, G is right adjoint to F , and η and ε are the unit and the
counit of the adjunction, respectively. We say that (2.4) is a colax-lax adjunction
[1, §3.9.1] if F is comonoidal, G is monoidal and the natural isomorphism
HomC (F (V ), W ) ∼
= HomC (V, G(W ))
(V ∈ B, W ∈ C)
of the adjunction is an isomorphism of monoidal functors from B op × C to Set. This
notion is in fact an instance of doctrinal adjunctions [26] and therefore we have the
following result (see [1, §3.9.1] for details):
Lemma 2.4. Let hF, G, η, εi : B ⇀ C be an adjunction between monoidal categories
B and C. If F is comonoidal (respectively, G is monoidal), then there uniquely
exists a monoidal structure of G (respectively, a comonoidal structure of F ) such
that hF, G, η, εi is a colax-lax adjunction.
An adjoint functor is often determined only up to isomorphism. Thus we consider
the case where a functor F : B → C has two right adjoint functors G and G′ . Then
there is a canonical isomorphism G ∼
= G′ induced from
(2.5)
HomC (V, G(W )) ∼
= HomB (F (V ), W ) ∼
= HomC (V, G′ (W )).
If F is comonoidal, then both G and G′ are monoidal by Lemma 2.4. Since the
isomorphisms in (2.5) are monoidal, the canonical isomorphism G ∼
= G′ is in fact an
isomorphism of monoidal functors. Similarly, two left adjoint functors of a monoidal
functor are canonically isomorphic as comonoidal functors.
ON UNIMODULAR FINITE TENSOR CATEGORIES
7
Now suppose that B and C are rigid. For a functor T : B → C, we define T ! to
be the following composition of functors:
(−)∗
∗
T op
(−)
T ! : B −−−−−−→ B op −−−−−−→ C op −−−−−−→ C.
If F : B → C is strong monoidal, then there is an isomorphism F ! ∼
= F of monoidal
functors [30, Lemma 1.1]. If, moreover, L is left adjoint to F , then L! is right
adjoint to F [6, Lemma 3.5]. Indeed, we have isomorphisms
(2.6)
HomC (V, L! (W )) ∼
= HomC (L(W ∗ ), V ∗ )
∼
= HomC (W ∗ , F (V ∗ ))
∼
= HomC (F ! (V ), W ) ∼
= HomC (F (V ), W )
natural in the variables V ∈ C and W ∈ B. Similarly, if R is right adjoint to F ,
then R! is left adjoint to F .
By Lemma 2.4, L is a comonoidal functor. Hence L! is a monoidal functor with
monoidal structure given by ∗ L0 : 1 → L! (1) and
∗
L2 (X,Y )
L! (X) ⊗ L! (Y ) = ∗ (L(Y ∗ ) ⊗ L(X ∗ )) −−−−−−−−−−→ ∗ L(Y ∗ ⊗ X ∗ ) = L! (X ⊗ Y ),
where L0 and L2 are the comonoidal structure of L. On the other hand, since L! is
right adjoint to F , it has another monoidal structure by Lemma 2.4. The following
lemma says that these two structures are the same.
Lemma 2.5. Let F : B → C a strong monoidal functor between rigid monoidal
categories. Suppose that F has a left adjoint L and a right adjoint R. Then the
canonical isomorphism L! ∼
= R is an isomorphism of monoidal functors.
Proof. The isomorphism HomB (F (V ), W ) ∼
= HomC (V, L! (W )) obtained in the above
is in fact an isomorphism of monoidal functors. Hence
HomC (V, R(W )) ∼
= HomC (V, L! (W ))
= HomB (F (V ), W ) ∼
as monoidal functors from C op × B to Set. Now the result follows from the Yoneda
lemma.
Applying Lemma 2.5 to the functor F rev : B rev → C rev induced by F , we also
have an isomorphism R ∼
= ! L of monoidal functors, where ! L = L(∗ −)∗ .
Since R is monoidal, A = R(1) is an algebra in C as the image of the trivial
algebra 1 ∈ C. Similarly, since L is comonoidal, C = L(1) is a coalgebra in C. The
above lemma implies that A ∼
= ∗ C as algebras in C.
2.5. Hopf monads. Let T = (T, µ, η) be a monad [28, VI.1] on a category C with
multiplication µ and unit η. By a T -module, we mean an object M ∈ C endowed
with a morphism ρM : T (M ) → M satisfying
ρM ◦ µM = ρM ◦ T (ρM ) and
ρM ◦ ηM = idM .
This notion is also called a “T -algebra” in literature but we do not use this term in
this paper. We denote by T C the category of T -modules (= the Eilenberg-Moore
category of T -algebras [28, VI.2]).
Let C be a monoidal category. A bimonad [5, 4] on C is a monad T = (T, µ, η)
on C such that the functor T is comonoidal and the natural transformations µ and
8
KENICHI SHIMIZU
η are comonoidal natural transformations. If M and N are T -modules, then their
tensor product M ⊗ N is also a T -module by
ρM ⊗ρN
T
2
−→ T (M ) ⊗ T (N ) −−−−−−−−−→ M ⊗ N,
T (M ⊗ N ) −−−−
where ρM and ρN are the action of T on M and N , respectively. The category T C
of T -modules is a monoidal category with this operation.
Now suppose that C is rigid. Then a Hopf monad on C is a bimonad T on C
endowed with natural transformations
SV : T (T (V )∗ ) → V ∗
and S V : T (∗ T (V )) → ∗ V
(V ∈ C)
satisfying certain conditions. The natural transformations S and S are called the
left antipode and the right antipode of T , respectively. If T is a Hopf monad on C,
then the monoidal category T C is rigid. The left dual object of M ∈ T C is the left
dual object M ∗ in C with the action given by
(2.7)
T (ρ∗ )
S
M
→ M ∗.
T (M ∗ ) −−−−M−→ T (T (M )∗ ) −−−
2.6. Finite abelian categories. Let k be a field. Given a k-algebra A, we denote
by A-mod and mod-A the categories of finite-dimensional left and right A-modules,
respectively. The following variant of the Eilenberg-Watts theorem [14, 37] will be
used extensively:
Lemma 2.6. Let A and B be finite-dimensional k-algebras. For a k-linear functor
F : mod-A → mod-B, the following three assertions are equivalent:
(1) F is left exact.
(2) F has a left adjoint.
(3) F ∼
= HomA (M, −) for some finite-dimensional B-A-bimodule M .
The following three assertions are also equivalent:
(1)′ F is right exact.
(2)′ F has a right adjoint.
(3)′ F ∼
= (−) ⊗A M for some finite-dimensional A-B-bimodule M .
By a finite abelian category over k, we mean a k-linear abelian category equivalent
to mod-A for some finite-dimensional k-algebra A.
Lemma 2.7. Let A be a finite abelian category over k, and let T be a k-linear right
exact monad on A. Then T A is also a finite abelian category over k.
Proof. Following Eilenberg and Moore [15, Proposition 5.3], the category T A is a
k-linear abelian category such that the forgetful functor F : T A → A preserves and
reflects exact sequences. To complete the proof, it is enough to show that T A has
a projective generator. Let L be a left adjoint of F , and let P ∈ A be a projective
generator. Then Q = L(P ) ∈ T A is projective, since
∼ HomA (P, F (−)) = HomA (P, −) ◦ F
HomT (L(P ), −) =
is exact. Now let M ∈ T A. Then there exists an epimorphism f : P ⊕m → F (M )
in A. Note that L preserves epimorphisms as it is left adjoint. Since U is faithful,
the counit ε of the adjunction is epic [28, IV.3]. Hence the composition
L(f )
ε
Q⊕m = L(P ⊕m ) −−−−→ LU (M ) −−−−→ M
is epic. Therefore Q ∈ T A is a projective generator.
ON UNIMODULAR FINITE TENSOR CATEGORIES
9
2.7. Finite tensor categories. Following [19], a finite tensor category over k is a
rigid monoidal category C such that C is a non-zero finite abelian category over k
and the tensor product ⊗ : C × C → C is k-linear in each variable.
We note that the tensor product of a finite tensor category C is exact in each
variable [2, Proposition 2.18], since there are adjunctions
V ∗ ⊗ (−) ⊣ V ⊗ (−) ⊣
∗
V ⊗ (−) and
(−) ⊗ ∗ V ⊣ (−) ⊗ V ⊣ (−) ⊗ V ∗
for each V ∈ C, where F ⊣ G means that F is a left adjoint functor of G.
Unlike [19], and like [13], we do not assume that the unit object 1 ∈ C is a simple
object (thus our finite tensor category is in fact a finite multi-tensor category in the
sense of [19]). It is known that 1 ∈ C can be written as the direct sum
1 = 11 ⊕ · · · ⊕ 1m
of pairwise non-isomorphic simple objects 11 , . . . , 1m ∈ C such that
(2.9)
1i ⊗ 1i ∼
= 1i , 1i ⊗ 1j = 0 (i 6= j), and 1∗i ∼
= 1i
for all i, j = 1, . . . , m. In particular, 1 is a semisimple object. Thus:
Lemma 2.8. The unit object 1 ∈ C is a simple object if EndC (1) ∼
= k.
The full subcategories Cij := 1i ⊗ C ⊗ 1j ⊂ C are called the component subcat-
(2.8)
egories of C. Now suppose that X ∈ Cpq and Y ∈ Crs . If q 6= r, then X ⊗ Y = 0
by (2.9). Otherwise, we have an inequality
(2.10)
ℓ(X ⊗ Y ) ≥ ℓ(X) · ℓ(Y ),
where ℓ(M ) denotes the length of M . In particular, X ⊗ Y 6= 0 if X ∈ Cpq and
Y ∈ Cqr are non-zero objects; see [16] for more details.
2.8. Finite module categories. Let C be a monoidal category. A left C-module
category is a category M endowed with a functor = : C × M → M, called the
action of C, and natural isomorphisms
1=M ∼
= M and (X ⊗ Y ) = M ∼
= X = (Y = M ) (X, Y ∈ C, M ∈ M)
satisfying the axioms similar to those for monoidal categories. See [31] for the
precise definitions of a left C-module category and related notions.
Now suppose that C is a finite tensor category over a field k. We say that a left
C-module category M is finite if its underlying category is a finite abelian category
over k and the action = : C × M → M of C is k-linear in each variable and right
exact in the first variable. Note that the action = is always exact in the second
variable since, for each V ∈ C, there are adjunctions
V ∗ = (−)
⊣
V = (−)
⊣
∗
V = (−).
Let M be a finite left C-module category. For M ∈ M, the functor (−) = M has
a right adjoint by Lemma 2.6. We denote it by Hom(M, −). By definition, there is
an isomorphism of vector spaces
(2.11)
HomC (V, Hom(M, N )) ∼
= HomM (V = M, N )
natural in the variables V and N . The assignment (M, N ) 7→ Hom(M, N ) uniquely
extends to a functor Hom : Mop × M → C, called the internal Hom functor for M,
in such a way that (2.11) is natural also in the variables M and N .
The counit of the adjunction (−) = M ⊣ Hom(M, −), denoted by
(2.12)
evM,N : Hom(M, N ) = M → N
(N ∈ M),
10
KENICHI SHIMIZU
is often called the evaluation. For L, M, N ∈ M, the composition
(2.13)
◦L,M,N : Hom(M, N ) ⊗ Hom(L, M ) → Hom(L, N )
is defined to be the morphism corresponding to
(Hom(M, N ) ⊗ Hom(L, M )) = L
id=evL,M
evM,N
∼
= Hom(M, N ) = (Hom(L, M ) = L) −−−−−−−→ Hom(M, N ) = M −−−−−→ N
via natural isomorphism (2.11), and the identity
(2.14)
idM : 1 → End(M ) (= Hom(M, M ))
is the morphism corresponding to the isomorphism 1 = M ∼
= M via (2.11). The
composition and the identity behave like those in a usual category (i.e., M is a
C-enriched category). In particular, End(M ) is an algebra in C.
Example 2.9. Set V = mod-k. Every finite abelian category M over k has a natural
structure of a finite left V-module category with action “·” determined by
∼ Homk (V, HomM (M, N )) (V ∈ V, M, N ∈ M).
HomA (V · M, N ) =
By definition, Hom(M, N ) = HomM (M, N ) for all M, N ∈ M. In this example,
(2.12), (2.13) and (2.14) coincide with the evaluation, the composition of maps and
the identity map, respectively.
Example 2.10. Let B and C be a finite tensor categories, and let F : B → C be
a k-linear right exact strong monoidal functor. Then C is a finite left B-module
category with action given by X = V = F (X) ⊗ V (X ∈ B, V ∈ C). By Lemma 2.6,
F has a right adjoint functor R. Since
HomC (X = V, W ) ∼
= HomC (F (X), W ⊗ V ∗ ) ∼
= HomB (X, R(W ⊗ V ∗ ),
the internal Hom functor is given by Hom(V, W ) = R(W ⊗ V ∗ ).
Example 2.11. Let A be an algebra in a finite tensor category C. The category CA
of right A-modules in C has a natural structure of a finite left C-module category
with action given by X = M = X ⊗ M for X ∈ C and M ∈ CA . We have
Hom(M, N ) = (M ⊗A ∗ N )∗
(M, N ∈ CA ),
where ⊗A is the tensor product over A [31, Example 2.10.8].
We go back to the general setting. Let M be a finite module category over a
finite tensor category C. Following [31], there is a natural isomorphism
(2.15) Hom(X = M, Y = N ) ∼
= Y ⊗ Hom(M, N ) ⊗ X ∗ (M, N ∈ M, X, Y ∈ C).
For M ∈ M, the monad T associated to (2.11) is given by
T = Hom(M, (−) = M ) ∼
= (−) ⊗ Hom(M, M ) = (−) ⊗ A,
where A = End(M ). Thus a T -module is precisely a right A-module in C, and the
comparison functor [28, VI.3] for (2.11) is given by
K M : M → CA ,
N 7→ Hom(M, N ) (N ∈ M),
where the action of A on Hom(M, N ) is given by (2.13) with L = M . By (2.15), the
functor KM is in fact a functor of left C-module categories. Applying the Barr-Beck
theorem [28, VI.7] to KM , we obtain the following theorem:
Theorem 2.12 ([16]). The functor KM above is an equivalence of left C-module
categories if and only if the following two conditions are satisfied:
ON UNIMODULAR FINITE TENSOR CATEGORIES
11
(K1) The functor Hom(M, −) : M → C is exact.
(K2) Every object of M is a quotient of an object of the form X = M , X ∈ C.
See also [12], where a detailed proof of this theorem is given. They also studied
conditions equivalent to (K1) and (K2). For later use, we note from [12] that the
condition (K2) is equivalent to that Hom(M, −) is faithful.
3. The central Hopf monad
3.1. Ends and coends. Let A and B be categories, and let S and T be functors
..
from Aop × A to B. A dinatural transformation ξ : S −−→ T is a family
ξ = {ξX : S(X, X) → T (X, X)}X∈A
of morphisms in B parametrized by the objects of A such that
T (idX , f ) ◦ ξX ◦ S(f, idX ) = T (f, idY ) ◦ ξY ◦ S(idY , f )
for all morphism f : X → Y in A. We regard an object X ∈ B as a functor from
Aop × A to B sending all morphisms to idX . An end of S is a pair (E, p) consisting
..
of an object E ∈ B and a dinatural transformation p : E −−→ S satisfying a certain
universal property (by abuse of terminology, we also refer to E and p as an end). A
coend of T is a pair (C, i) consisting of an object C ∈ B and a ‘universal’ dinatural
..
transformation i : T −−→ C. The universal property ensures that a (co)end is
unique up to isomorphism if it exists. An end (E, p) of S and a coend (C, i) of T
will be denoted, respectively, by
Z X∈A
Z
T (X, X).
S(X, X) and C =
E=
X∈A
We refer the reader to [28] for general treatments of (co)ends. For reader’s convenience, we here collect some formulas for (co)ends. Suppose that A is essentially
small. Given two functors F1 , F2 : A → B, we denote by Nat(F1 , F2 ) the set of
natural transformations from F1 to F2 . Then
..
pX : Nat(F1 , F2 ) −−→ HomB (F1 (X), F2 (X)),
α 7→ αX
(X ∈ A)
is an end of HomB (F1 (−), F2 (−)). With integral notation, we have
Z
(3.1)
Nat(F1 , F2 ) =
HomB (F1 (X), F2 (X)).
X∈A
The following formula will be used extensively: If a coend of T exists, then
Z
Z X∈A
HomB (T (X, X), V ) (V ∈ B).
T (X, X), V ) =
(3.2)
HomB (
X∈A
Since the category Set is complete, the end of the right-hand side of (3.2) exists
without the assumption that a coend of T exists. Thus, by the parameter theorem
for ends [28, IX.7], we obtain a functor
Z
♮
HomB (T (X, X), V ) (V ∈ B).
T : B → Set, V 7→
X∈A
Lemma 3.1. A coend of T exists if and only if T ♮ is representable.
12
KENICHI SHIMIZU
Proof. In view of (3.2), it is sufficient to show that a coend of T exists if T ♮ is
representable. Let C be an object representing T ♮ . By the definition of the functor
T ♯ , there exists a family
{φX,V : HomB (C, V ) → HomB (T (X, X), V )}X∈A,V ∈B
of maps that is natural in V and dinatural in X. By the Yoneda lemma, φX,V is
induced by a morphism iX : T (X, X) → C. The family i = {iX } is dinatural in X,
and the pair (C, i) is indeed a coend of T .
3.2. The central Hopf monad. Suppose that C is a rigid monoidal category such
that the coend
Z X∈C
(3.3)
Z(V ) =
X∗ ⊗ V ⊗ X
exists for all V ∈ C. Then the assignment V 7→ Z(V ) extends to an endofunctor Z
on C. Day and Street [10] showed that the functor Z has a structure of a monad
such that Z C ∼
= Z(C) as categories. Following Bruguières and Virelizier [6], the
monad Z has a structure of a quasitriangular Hopf monad and the isomorphism
∼
Z C = Z(C) is in fact an isomorphism of braided monoidal categories. We call the
Hopf monad Z the central Hopf monad on C.
For later use, we recall from [10] and [6] the definition of the central Hopf monad
and the construction of the isomorphism Z C ∼
= Z(C). For V, X ∈ C, we denote by
iV (X) : X ∗ ⊗ V ⊗ X → Z(V ) the component of the universal dinatural transformation. Then the comonoidal structure
Z0 : Z(1) → 1 and Z2 (V, W ) : Z(V ⊗ W ) → Z(V ) ⊗ Z(W ) (V, W ∈ C)
are defined to be the unique morphisms such that Z0 ◦ i1 (X) = evX and
Z2 (V, W ) ◦ iV ⊗W (X) = (iV (X) ⊗ iW (X)) ◦ (idX ∗ ⊗ idV ⊗ coevX ⊗ idW ⊗ idX )
for all X ∈ C, respectively. The unit of Z is given by ηV = iV (1) (V ∈ C). To
define the multiplication of Z, we note that
(2)
iV (X, Y ) := iZ(V ) (Y ) ◦ (idY ∗ ⊗ iV (X) ⊗ idY )
(X, Y ∈ C)
X2∗ ⊗ Y2∗
is a coend of (X1 , Y1 , X2 , Y2 ) 7→
⊗ V ⊗ X1 ⊗ Y1 (X1 , X2 , Y1 , Y2 ∈ C) by the
Fubini theorem for coends [28, IX.8]. Hence we can define µ : Z 2 → Z by
(3.4)
(2)
µV ◦ iV (X, Y ) = iV (X ⊗ Y ) (V, X, Y ∈ C).
The left antipode of Z is given in Remark 3.2 below. We omit the descriptions of
the right antipode and the universal R-matrix of Z since we will not use them.
Following [10], we establish an isomorphism Z C ∼
= Z(C) of categories. We first
note that, by (3.1) and (3.2), there are natural isomorphisms
R
HomC (Z(V ), W ) ∼
= X∈C HomC (X ∗ ⊗ V ⊗ X, W )
R
∼
= X∈C HomC (V ⊗ X, X ⊗ W )
∼
= Nat(V ⊗ (−), (−) ⊗ W )
for V, W ∈ C. Let ∂V (−) : V ⊗(−) → (−)⊗Z(V ) denote the natural transformation
corresponding to idZ(V ) via the above chain of isomorphisms. If V is a Z-module
with action ρ, then one can check that
∂V (X)
idX ⊗ρ
σV (X) : V ⊗ X −−−−−−−−→ X ⊗ Z(V ) −−−−−−−−→ X ⊗ V
(X ∈ C)
ON UNIMODULAR FINITE TENSOR CATEGORIES
13
is a half-braiding for V . This construction gives rise to an isomorphism Z C ∼
= Z(C)
of categories. The isomorphism is in fact monoidal and commutes with the forgetful
functors to C.
Remark 3.2. The Hopf monad Z can be defined by using the natural transformation
∂ instead of the dinatural transformation i (in fact, this is the way of [6]). By using
∂, the left antipode is defined by
(3.5)
(idX ⊗ SV ) ◦ ∂Z(V )∗ (X) = ∂V (∗ X)∗
(V, X ∈ C).
3.3. Existence of coends. To apply the above Hopf monadic description of the
center to finite tensor categories, we show that a coend of certain type of functors,
including (3.3), exists in a finite tensor category over a field k.
Given k-linear abelian categories A1 , . . . , An and C, we denote by
Lexn (A1 , . . . , An ; C) (respectively, Rexn (A1 , . . . , An ; C))
the category of functors from A1 × · · · × An to C being k-linear left exact (respectively, right exact) in each variable. For simplicity, we write
Lex(A, C) = Lex1 (A; C) and
Rex(A, C) = Rex1 (A; C).
A tensor product [11, §5] of k-linear abelian categories A1 , . . . , An is a k-linear
abelian category T endowed with ⊠ ∈ Rexn (A1 , . . . , An ; T ) such that
Rex(T , C) → Rexn (A1 , . . . , An ; C) F 7→ F ◦ ⊠ (F ∈ Rex(T , C))
is an equivalence for any k-linear abelian category C. A tensor product of A1 , . . . , An
does not always exist; see [20]. If it exists, it is unique up to equivalence and is
denoted by A1 ⊠ · · · ⊠ An .
If A = mod-A and B = mod-B for some finite-dimensional k-algebras A and B,
then mod-(A ⊗k B) is a tensor product of A and B with
⊠ : A × B → mod-(A ⊗k B),
(X, Y ) 7→ X ⊗k Y
(X ∈ A, Y ∈ B)
[11, Proposition 5.3]. The following lemma is obtained by using this realization of
a tensor product of finite abelian categories:
Lemma 3.3 (Deligne [11, Proposition 5.13]). Let A and B be finite abelian categories over a field k. Then the following statements hold:
(1) A tensor product A ⊠ B exists and is a finite abelian category over k.
(2) The functor ⊠ : A × B → A ⊠ B is k-linear and exact in each variable.
(3) The functor Lex(A⊠ B, C) → Lex2 (A, B; C) induced by ⊠ is an equivalence
of categories for any k-linear abelian category C.
(4) There is a natural isomorphism
HomA⊠B (V ⊠ W, X ⊠ Y ) ∼
= HomA (V, X) ⊗k HomB (W, Y )
for V, X ∈ A and W, Y ∈ B.
Now let A and B be finite abelian categories over k. We consider the functor
Φ1 : A × B op × B → A,
(V, X, Y ) 7→ HomB (X, Y ) · V
(V ∈ A, X, Y ∈ B),
where “·” is the canonical action of mod-k on A given in Example 2.9. By Part (3)
of the above lemma, this functor induces a k-linear left exact functor
Φ2 : A ⊠ B op ⊠ B → A,
V ⊠ X ⊠ Y 7→ Φ1 (V, X, Y ).
14
KENICHI SHIMIZU
By Part (2) of the above lemma, we have a functor
Φ3 : A ⊠ B op → Lex(B, A),
M 7→ (V 7→ Φ2 (M ⊠ V )).
For simplicity, we express the functor Φ3 obtained in this way as
(3.6)
A ⊠ B op → Lex(B, A),
V ⊠ W 7→ HomB (W, −) · V
(V ∈ A, W ∈ B)
Lemma 3.4. The functor (3.6) is an equivalence.
Proof. We may assume that A = mod-A and B = mod-B for some finite-dimensional
algebras A and B. By Lemma 2.6 and the Yoneda lemma, we see that the following
functor is an equivalence:
(A-mod-B)op → L := Lex(A, B),
M 7→ HomB (M, −) (M ∈ A-mod-B),
where A-mod-B is the category of finite-dimensional A-B-bimodules. In view of
the above realization of a tensor product, we also have an equivalence
A ⊠ B op → (A-mod-B)op ,
V ⊠ W 7→ V ∗ ⊗k W
(V ∈ A, W ∈ B),
where A acts on V ∗ := Homk (V, k) by a · f = f (− · a) (a ∈ A, f ∈ V ∗ ). One can
check that (3.6) is obtained by composing these equivalences.
The following description of a quasi-inverse of (3.6) is important:
Lemma 3.5. Notations are the same as in Lemma 3.4. For all F ∈ Lex(B, A), a
coend of the functor
(3.7)
B × B op → A ⊠ B op ,
(X, Y ) 7→ F (X) ⊠ Y
exists. A quasi-inverse of (3.6) is given by
Z X∈B
Lex(B, A) → A ⊠ B op , F 7→
F (X) ⊠ X
(X, Y ∈ B)
(F ∈ Lex(B, A)).
Proof. For F ∈ Lex(B, A), there are isomorphisms
HomA⊠Bop (F (X) ⊠ Y, V ⊠ W ) ∼
= HomA (F (X), V ) ⊗k HomBop (Y, W )
∼
= HomA (F (X), HomB (W, Y ) · V )
natural in V ∈ A and W, X, Y ∈ B by Lemma 3.3 (4) and (2.15). Since both sides
are k-linear and left exact in the variables V and W , we obtain
HomA⊠Bop (F (X) ⊠ Y, M ) ∼
= HomA (F (X), Φ(M )(Y ))
(M ∈ A ⊠ B op ),
where Φ is the equivalence given by (3.6). Taking ends, we get
Z
HomA⊠Bop (F (X) ⊠ X, M ) ∼
= Nat(F, Φ(M )).
X∈A
Let Φ be a quasi-inverse of Φ. Since Nat(F, Φ(−)) is represented by Φ(F ), a coend
of (3.7) exists and is isomorphic to Φ(F ) by Lemma 3.1.
Following Kerler and Lyubashenko [27, §5.1.3], a coend of Q : A×Aop → B exists
if Q is k-linear and exact in each variable. Thus, in the case where F is exact, the
existence of a coend of (3.7) follows from their result. Theorem 3.6 below also
follows from their result in such a case.
ON UNIMODULAR FINITE TENSOR CATEGORIES
15
Theorem 3.6. Let C be a finite tensor category over a field k. Then coends
Z X∈C
Z X∈C
F (X ∗ ) ⊗ X
F (X ∗ ) ⊠ X and
exist for all F ∈ Lex(C, C).
Proof. Note that F (−∗ ) : C op → C is k-linear left exact if F ∈ Lex(C, C). Hence,
applying the above lemma to F (−∗ ), we see that the first coend exists. The second
coend is obtained by applying the right exact functor X ⊠ Y 7→ X ⊗ Y to the first
coend.
Remark 3.7. For F ∈ Lex(C, C), there is an isomorphism
Z X∈C
Z X∈C
F (X ∗ ) ⊠ X ∼
F (X) ⊠ ∗X.
=
Indeed, for every object C ∈ C ⊠ C op , the map
Dinat(F (−) ⊠ ∗ (−), C) → Dinat(F (−∗ ) ⊠ (−), C),
{iV }V ∈C 7→ {iV ∗ }V ∈C
is a bijection, where Dinat(P, Q) means the set of dinatural transformations from
P to Q. Similarly, there is an isomorphism
Z X∈C
Z X∈C
F (X) ⊗ ∗X.
F (X ∗ ) ⊗ X ∼
=
3.4. The center of a finite tensor category. Applying Theorem 3.6 to F =
(−) ⊗ V , we see that the coend in the right-hand side of (3.3) always exists in a
finite tensor category. As an application of this result, we prove:
Theorem 3.8. The center of a finite tensor category is a finite tensor category.
Proof. Let C be a finite tensor category over a field k. As we have remarked, the
central Hopf monad Z on C exists and therefore we can identify Z(C) as the category
!
∗
∗
Z C of Z-modules. Set Z (V ) = Z(V ) for V ∈ C. By Remark 3.7, we have
HomC (W, Z ! (V )) ∼
= HomC (Z(V ∗ ), W ∗ )
R
∼
= X∈C HomC (X ∗ ⊗ V ∗ ⊗ X, W ∗ )
R
∼
= X∈C HomC (X ⊗ W ⊗ ∗ X, V )
∼
= HomC (Z(W ), V )
for all V, W ∈ C. Hence the functor Z ! is a right adjoint of Z (this result is a special
case of [5, Corollary 3.12]). Now the result follows from Lemma 2.7.
Remark 3.9. Let C and D be finite tensor categories over a field k. Then C ⊠ D is
a k-linear monoidal category with tensor product determined by
(V ⊠ W ) ⊗ (X ⊠ Y ) = (V ⊗ X) ⊠ (W ⊗ Y ) (V, X ∈ C, W, Y ∈ D)
and unit 1 ⊠ 1. Following Deligne [11, Proposition 5.17], C ⊠ D is a finite tensor
category provided that k is a perfect field.
Theorem 3.8 is proved in [19] under the assumption that k is algebraically closed
and 1 ∈ C is simple. Their proof relies on the fact that C ⊠ C rev is a finite tensor
category and thus cannot be applied to our case.
16
KENICHI SHIMIZU
4. Characterizations of unimodularity
4.1. General assumptions. Let C be a finite tensor category over a field k. Then
C ⊠ C rev is a monoidal category with tensor product
(V ⊠ W ) ⊗ (X ⊠ Y ) = (V ⊗ X) ⊠ (Y ⊗ W ) (V, W, X, Y ∈ C)
and unit
(4.1)
1 ⊠ 1. Throughout this section, we assume that
C env := (C ⊠ C rev , ⊗, 1 ⊠ 1) is a finite tensor category,
which holds if the field k is perfect (see Remark 3.9). We note that (4.1) holds also in
the case where C is the representation category of a finite-dimensional (quasi-)Hopf
algebra.
4.2. The definition of unimodularity. Following [17], we recall the definition of
the distinguished invertible object and the unimodularity of a finite tensor category.
The category C is a finite C env -module category with the action determined by
(4.2)
(V ⊠ W ) = X = V ⊗ X ⊗ W
(V, W, X ∈ C).
Now we set A = Hom(1, 1) ∈ C env . By (2.15), we have
(4.3)
Hom(V ⊗ W, X ⊗ Y ) ∼
= (X ⊠ Y ) ⊗ A ⊗ (V ⊠ W )∗
(V, W, X, Y ∈ C).
Theorem 2.12 implies that the functor
(4.4)
C → (C env )A ,
V 7→ Hom(1, V ) ∼
= (V ⊠ 1) ⊗ AA
(V ∈ C)
is an equivalence of C env -module categories. In view of this equivalence, there exists
an object D ∈ C, which is unique up to isomorphism, such that
(4.5)
(D ⊠ 1) ⊗ AA ∼
= (A A)∗ .
Definition 4.1 ([17]). The object D is called the distinguished invertible object of
C, and the finite tensor category C is said to be unimodular if D ∼
= 1.
As its name suggests, D is an invertible object, i.e., the evaluation evD and the
coevaluation coevD are isomorphisms. In [17], the invertibility is proved under the
assumption that k is algebraically closed and 1 ∈ C is simple. Their proof relies on
the theory of the Frobenius-Perron dimension, and thus cannot be applied to our
case. To prove the invertibility of D, we first note:
Lemma 4.2. Hom(V, X)∗ ∼
= Hom(X, V ∗∗ ⊗ D).
Proof. By (4.3) with W = Y = 1 and (4.5), we compute
Hom(V, X)∗ ∼
= ((X ⊠ 1) ⊗ A ⊗ (V ⊠ 1)∗ )∗
∼
= (V ∗∗ ⊠ 1) ⊗ (D ⊠ 1) ⊗ A ⊗ (X ⊠ 1)∗
∼
= Hom(X, V ∗∗ ⊗ D).
Lemma 4.3. D is invertible.
Proof. By the previous lemma, we have natural isomorphisms
Hom(∗∗ V, 1)∗∗ ∼
= Hom(1, V ⊗ D)∗ ∼
= Hom(V ⊗ D, D)
ON UNIMODULAR FINITE TENSOR CATEGORIES
17
for V ∈ C. Thus we compute:
HomC (V, D ⊗ D∗ ) ∼
= HomC (V ⊗ D, D)
∼
= HomC env (1 ⊠ 1, Hom(V ⊗ D, D))
∼ HomC env (1 ⊠ 1, Hom(∗∗ V, 1)∗∗ )
=
∼
= HomC env (1 ⊠ 1, Hom(∗∗ V, 1))
∼
= HomC (V, 1).
= HomC (∗∗ V, 1) ∼
By the Yoneda lemma, D ⊗ D∗ ∼
= 1. This implies the invertibility of D, since C is
a multi-ring category in the sense of [16]; see [16, §1.15].
4.3. The algebra A as a coend. The first step for the proof of our main theorem
is to describe the algebra A as a coend of a certain functor. Note that the left
duality functor is an equivalence (−)∗ : C rev → C op with quasi-inverse ∗ (−). Hence,
by Lemmas 3.4 and 3.5, the functor
(4.6)
Φ : C env → Lex(C) (:= Lex(C, C)),
V ⊠ W 7→ HomC (W ∗ , −) · V
is an equivalence of categories with quasi-inverse given by
Z X∈C
env
F (X) ⊠ ∗X.
Φ : Lex(C) → C , F 7→
(4.7)
Recall that the category C is a finite C env -module category by (4.2). The internal
Hom functor for C is given as follows:
Lemma 4.4. Hom(V, W ) = Φ(W ⊗ (−) ⊗ V ∗ ).
Proof. For V, W ∈ C and F ∈ Lex(C), we compute
HomC env (Φ(F ), Φ(W ⊗ (−) ⊗ V ∗ )) ∼
= Nat(F, W ⊗ (−) ⊗ V ∗ )
R
∼
= X∈C HomC (F (X), W ⊗ X ⊗ V ∗ )
R
∼
= X∈C HomC ((F (X) ⊠ ∗ X) = V, W )
∼
= HomC (Φ(F ) = V, W ).
Since Φ is an equivalence, the claim follows from the above computation.
Let F ∈ Lex(C) and V, W ∈ C. We pay attention to the bijection
Nat(F, W ⊗ (−) ⊗ V ∗ ) ∼
= HomC (Φ(F ) = V, W )
in the proof of Lemma 4.4. The morphism f : Φ(F ) = V → W corresponding to a
natural transformation α : F → W ⊗ (−) ⊗ V ∗ via the above bijection is uniquely
determined by the property that the diagram
(F (X) ⊠ ∗X) = V
′
jF
(X)=V
/ Φ(F ) = V
f
/W
O
W ⊗evV ⊗∗X
F (X) ⊗ V ⊗ ∗X
αX ⊗V ⊗∗X
/ W ⊗ X ⊗ V ∗ ⊗ V ⊗ ∗X
commutes for all X ∈ C, where jF′ (X) : F (X) ⊠ ∗X → Φ(F ) is the component
of the universal dinatural transformation. In particular, the evaluation evV,W for
18
KENICHI SHIMIZU
V, W ∈ C is the morphism making the diagram
(4.8)
((W ⊗ X ⊗ V ∗ ) ⊠ ∗ X) = V
′′
jV,W
(X)=V
/ Hom(V, W ) = V
evV,W
W ⊗ X ⊗ V ∗ ⊗ V ⊗ ∗X
/W
W ⊗evV ⊗∗ X
′′
commutes for all X ∈ C, where jV,W
= jF′ with F = W ⊗ (−) ⊗ V ∗ . Now we
′′
set j = j1,1 . The algebra structure of A = Hom(1, 1) is described by using the
dinatural transformation j as follows:
Lemma 4.5. With the above notation, the multiplication m : A ⊗ A → A is a
unique morphism such that the diagram
j(X)⊗j(Y )
A⊗A o
(4.9)
(X ⊠ ∗X) ⊗ (Y ⊠ ∗ Y )
m
Ao
j(X⊗Y )
(X ⊗ Y ) ⊠ (∗ Y ⊗ ∗X)
commutes for all X, Y ∈ C. The unit u : 1 ⊠ 1 → A is given by u = j(1).
Proof. It is easy to see that the unit of A is given as stated. For X, Y ∈ C, we have
a commutative diagram
((X ⊠ ∗X) ⊗ (Y ⊠ ∗ Y )) = 1
(j(X)⊗j(Y ))=1
/ (A ⊗ A) = 1
∼
=
m=1
/ A=1
∼
=
(X ⊠ ∗X) = ((Y ⊠ ∗ Y ) = 1)
j(X)=(j(Y )=1)
(X⊠∗X)=ev∗ Y
/ A = (A = 1)
ev1,1
A=ev1,1
(X ⊠ ∗ X) = 1
j(X)=1
/ A=1
/
ev1,1
1
by (4.8) and the definition of m. Again by (4.8), the composition along the bottom
row is ev∗X . Hence we obtain:
ev1,1 ◦ (m = 1) ◦ ((j(X) ⊗ j(Y )) = 1) = ev∗X ◦ (idX⊠∗X = ev∗ Y )
= ev∗X ◦ (idX ⊗ ev∗ Y ⊗ id∗X )
= ev∗ Y ⊗∗ X
= ev1,1 ◦ (j(X ⊗ Y ) = 1).
Since the map HomC env (M, A) → HomC (M = 1, 1) given by f 7→ ev1,1 ◦ (f = 1) is
bijective, the commutativity of (4.9) follows.
4.4. The algebra A and the central Hopf monad. For V, X ∈ C, we set
Z(V ) = A = V
and
iV (X) = j(X ∗ ) = V : X ∗ ⊗ V ⊗ X → Z(V ),
where A and j are as before. Since A is an algebra in C env , the functor Z has a
structure of a monad. More precisely, the multiplication of Z is given by
m=V
µV : Z 2 (V ) = A = (A = V ) = (A ⊗ A) = V −−−−−→ A = V = Z(V )
(V ∈ C)
ON UNIMODULAR FINITE TENSOR CATEGORIES
19
and the unit of Z is given by
ηV : V = 1 = V −−−−→ A = V = Z(V ) (V ∈ C).
u=V
Note that {iV (X)}X∈C is a coend of (X, Y ) 7→ X ∗ ⊗ V ⊗ Y . By Lemma 4.4, one
can check that the unit η is given by ηV = iV (1) for V ∈ C and the multiplication
µ is determined by the same formula as (3.4). In conclusion, the monad Z under
consideration is precisely the central Hopf monad on C.
Let K : C → H := (C env )A be the equivalence given by (4.4). Note that the
functor T = A ⊗ (−) defines a monad on H such that T H = A (C env )A . Since K is
in fact an equivalence of C env -module categories, it induces an equivalence between
Z C and T H. More precisely, if M is a Z-module with action ρ, then K(M ) ∈ H is
an A-bimodule with the left action given by
∼
=
K(ρ)
A ⊗ K(M ) −−−−−→ K(A = M ) = K(Z(M )) −−−−→ K(M )
and this construction gives rise to an equivalence of categories
≈
e : Z C −−
(4.10)
K
→ T H = A (C env )A , M 7→ K(M ) (M ∈ Z C).
e it is
Recall from §2.5 that Z C can be identified with Z(C). By the definition of K,
obvious that the following diagram commutes:
(4.11)
Z(C)
e
K
/ TH
A (C
env
)A
FA
U
C
K
/H
(C env )A ,
where U and FA are forgetful functors.
Remark 4.6. Etingof and Ostrik [19, Corollary 3.35] showed that A (C env )A is equivalent to Z(C). However, since they did not give an equivalence in an explicit way, it
is not clear that there exists a commutative diagram like (4.11). In this paper, we
give an equivalence A (C env )A ≈ Z(C) in a somewhat explicit way by investigating
the relation between the algebra A and the central Hopf monad. The commutativity
of (4.11) is obvious from our point of view.
4.5. Characterizations of unimodularity. In this subsection, we prove the main
theorem of this paper. Recall our assumption that C is a finite tensor category with
property (4.1). Let L and R be a left adjoint and a right adjoint of the forgetful
functor U : Z(C) → C, and let D ∈ C be the distinguished invertible object. The
key observation is the following lemma:
Lemma 4.7. There are natural isomorphisms
L(D ⊗ −) ∼
=R∼
= L(− ⊗ D) and R(D∗ ⊗ −) ∼
=L∼
= R(− ⊗ D∗ ).
e be the equivalence given by (4.10). By Lemma 2.1 and the commuProof. Let K
tativity of the diagram (4.11), we have
e
KL(V
)∼
= A A ⊗ K(V ) ∼
= A A ⊗ (V ⊠ 1) ⊗ AA .
Note that ∗ (AA ) ∼
= A A ⊗ (D ⊠ 1) by (4.5). Again by Lemma 2.1, we have
e
KR(V ) ∼
= ∗ (AA ) ⊗ (V ⊠ 1) ⊗ AA
∼
e
⊗V)
= KL(D
= A A ⊗ (D ⊠ 1) ⊗ (V ⊠ 1) ⊗ AA ∼
20
KENICHI SHIMIZU
for V ∈ C. Hence we obtain the first natural isomorphism. The third one is obtained
from the first one and the fact that D is invertible as follows:
R(D∗ ⊗ V ) ∼
= L(D ⊗ D∗ ⊗ V ) ∼
= L(V ).
For a functor T between rigid monoidal categories, we set T ! (X) = ∗ T (X ∗ ). Recall
from §2.4 that R ∼
= L! and L ∼
= R! . The second one is obtained from this fact and
the third natural isomorphism as follows:
R(V ) ∼
= ∗L(V ∗ ) ∼
= ∗R(D∗ ⊗ V ∗ ) ∼
= ∗R((V ⊗ D)∗ ) ∼
= L(V ⊗ D).
The last one is obtained from the second one and the invertibility of D.
The above lemma implies many relations between U , L, R and D. Here we prove
the following lemma, which describes a left adjoint of L and a right adjoint of R in
terms of U and D.
Corollary 4.8. There are natural isomorphisms
HomC (D ⊗ U (X), V ) ∼
= HomC (U (X) ⊗ D, V ),
= HomZ(C) (X, L(V )) ∼
HomC (V, D∗ ⊗ U (X)) ∼
= HomZ(C) (R(V ), X) ∼
= HomC (V, U (X) ⊗ D∗ ).
Proof. We only give the first natural isomorphism, since the others are obtained in
a similar way. By Lemma 4.7 and the invertibility of D, we have:
∼ HomZ(C) (X, R(D∗ ⊗ V ))
HomZ(C) (X, L(V )) =
∼
= HomZ(C) (U (X), D∗ ⊗ V )
∼
= HomZ(C) (D ⊗ U (X), V ).
Corollary 4.9. L and R are exact.
Now we prove our main theorem:
Theorem 4.10. With the above notation, the following assertions are equivalent:
(1) C is unimodular.
(2) U is a Frobenius functor, i.e., L ∼
= R.
(3) There is a natural isomorphism L(V ∗ ) ∼
= L(V )∗ .
∗ ∼
(4) There is a natural isomorphism R(V ) = R(V )∗ .
Moreover, if the unit object 1 ∈ C is simple, the above assertions are equivalent to
each of the following assertions:
(5) L(1) ∼
= L(1)∗ .
(6) R(1) ∼
= R(1)∗ .
(7) HomZ(C) (1, L(1)) 6= 0.
(8) HomZ(C) (R(1), 1) 6= 0.
Proof. It is obvious from Lemma 4.7 that (1) implies (2). Using the isomorphisms
R∼
= L! and L ∼
= R! , we easily see that (2), (3) and (4) are equivalent. We show
that (2) implies (1). If (2) holds, then
HomC (D, V ) ∼
= HomZ(C) (1, R(V )) ∼
= HomC (1, V )
= HomZ(C) (1, L(V )) ∼
by Corollary 4.8 with X = 1. Thus D ∼
= 1, i.e., C is unimodular. Hence we have
showed that the assertions (1), (2), (3) and (4) are equivalent.
It is obvious that (3) implies (5) (without the assumption that 1 ∈ C is a simple
object). If (5) holds, then we have
HomZ(C) (1, L(1)) ∼
= HomZ(C) (1, L(1)∗ ) ∼
= HomZ(C) (L(1), 1) ∼
= HomC (1, 1)
ON UNIMODULAR FINITE TENSOR CATEGORIES
21
and therefore (7) holds (again without the assumption on 1). We prove (4) ⇒ (6)
and (6) ⇒ (8) in a similar way. Moreover, since
∼ HomZ(C) (L(1)∗ , 1) ∼
HomZ(C) (1, L(1)) =
= HomZ(C) (R(1), 1),
the assertions (7) and (8) are equivalent.
Now we suppose that 1 ∈ C is simple. To complete the proof, it is sufficient to
show that (7) implies (1). If (7) holds, then we have
HomC (D, 1) ∼
= HomZ(C) (1, L(1)) 6= 0
by Corollary 4.8. Since 1 ∈ C is assumed to be simple, every invertible object of C
is also simple. Thus, by Schur’s lemma, we have D ∼
= 1, i.e., (1) holds.
5. Applications, I. Further results on the unimodularity
5.1. General assumptions. In this section, we apply our techniques to investigate
further properties of the distinguished invertible object. As before, C is a finite
tensor category over a field k with property (4.1), D ∈ C is the distinguished
invertible object, U : Z(C) → C is the forgetful functor, and L and R are a left
adjoint and a right adjoint of U , respectively.
5.2. The Radford S 4 -formula. Let S denote the left duality functor on C, and
let ID : C → C denote the monoidal functor defined by ID (X) = D ⊗ X ⊗ D∗
(X ∈ C). One of the main results of [17] is that there exists an isomorphism
(5.1)
S4 ∼
= ID
of monoidal functors (the Radford S 4 -formula). Here we explain how this formula
looks like from the argument in the previous section (see [12] and [34] for other
approaches to the Radford S 4 -formula).
We have used the equivalence Φ : C env → Lex(C) given by (4.6) to prove our
main theorem. As before, we denote its quasi-inverse by Φ. We compute
R X,Y ∈C
Φ((F (X) ⊠ ∗ X) ⊗ (G(X) ⊠ ∗ Y ))
Φ(Φ(F ) ⊗ Φ(G)) =
R X,Y ∈C
=
Φ((F (X) ⊗ G(Y )) ⊠ ∗ (X ⊗ Y ))
R X,Y ∈C
=
HomC (X ⊗ Y, −) · (F (X) ⊗ G(Y )).
Thus, F ⋆ G := Φ(Φ(F ) ⊗ Φ(G)) is the Day convolution [9] of F and G, and the
equivalence Φ is in fact a monoidal equivalence
Φ : (C env , ⊗, 1 ⊠ 1) → (Lex(C), ⋆, J),
where J = HomC (1, −) · 1. In particular, Φ sends an algebra in C env to an algebra
in Lex(C) with respect to the Day convolution, i.e., a k-linear left exact monoidal
endofunctor on C (see [9, Example 3.2.2]).
Now let Hom denote the internal Hom functor for the C env -module category C,
and let A = Hom(1, 1) be the algebra in C env used to define D. By Lemma 2.2 and
the definition of D, we obtain an isomorphism
(5.2)
A∗∗ ∼
= AD := D ⊗ A ⊗ D∗
of algebras in C env . Since (−)∗∗ : C env → C env is an equivalence, we have
Z X
Z X
Z X∈C
∗∗
∗∗∗
∗
∗∗ ∼
∗∗ ∼
∼
X ⊠ X=
X ∗∗∗∗ ⊠ ∗ X ∼
(X ⊠ X) =
A =
= Φ(S 4 )
22
KENICHI SHIMIZU
(cf. Remark 3.7). We also have AD ∼
= Φ(ID ) by Lemma 4.4. One can check that
these isomorphisms are in fact isomorphisms of algebras in C env . Thus, to prove the
Radford S 4 -formula (5.1), we only have to apply Φ to (5.2).
5.3. Faithfulness of adjoints. We have showed that L and R are exact functors
(Corollary 4.9). Here we discuss the faithfulness of L and R. The following lemma,
which is possibly well-known, is useful for our purpose:
Lemma 5.1. For an exact functor F : A → B between abelian categories, the
following assertions are equivalent:
(1) F is faithful.
(2) F reflects isomorphisms.
(3) F reflects zero objects.
Proof. For an object M of an abelian category, we denote by 0M the zero morphism
on M . Let X ∈ A be an object such that F (X) = 0. Then we have
F (0X ) = 0F (X) = idF (X) = F (idX ).
Thus, if (1) holds, then 0X = idX , and therefore X = 0. If (2) holds, then 0X is an
isomorphism since idF (X) is, and therefore X = 0. Summarizing, we have proved
that either of (1) or (2) implies (3).
Now suppose that (3) holds. If f is a morphism in A such that F (f ) = 0, then
we have F (Im(f )) = Im(F (f )) = 0 since F is exact. Thus Im(f ) = 0 and therefore
f = 0. This implies that (1) holds.
Similarly, if f is a morphism such that F (f ) is an isomorphism, then we have
F (Ker(f )) = Ker(F (f )) = 0 and F (Coker(f )) = Coker(F (f )) = 0 since F is exact.
Thus we have Ker(f ) = 0 and Coker(f ) = 0 and therefore f is an isomorphism.
This implies that (2) holds. The proof is done.
Theorem 5.2. We decompose the unit object 1 ∈ C as 1 = 11 ⊕ · · · ⊕ 1m as in
(2.8), and suppose that EndC (1i ) ∼
= k for all i = 1, . . . , m. Then the following
assertions are equivalent:
(1) L is faithful.
(2) R is faithful.
(3) The full subcategory Cij := 1i ⊗ C ⊗ 1j is zero whenever i 6= j.
Proof. The equivalence (1) ⇔ (2) follows from L ∼
= R! . Now let A be the algebra
env
in C
used to define the distinguished invertible object. By the argument in the
proof of Lemma 4.7, the faithfulness of L is equivalent to the faithfulness of
L′ : C → C env ,
V 7→ A ⊗ (V ⊠ 1) ⊗ A.
To show (1) ⇒ (3), we compute, for V ∈ Cij with i 6= j,
L′ (V ) ∼
= A ⊗ (1i ⊠ 1) ⊗ (V ⊠ 1) ⊗ (1j ⊠ 1) ⊗ A
∼
= A ⊗ (1 ⊠ 1i ) ⊗ (V ⊠ 1) ⊗ (1 ⊠ 1j ) ⊗ A = 0
by (2.9) and (4.3). Thus, by Lemma 5.1, L′ cannot be faithful if Cij 6= 0 for some
i 6= j. In other words, (1) implies (3).
Now we suppose that (3) holds. Then C = C11 ⊕ · · · ⊕ Cmm . By Lemma 5.1, it is
sufficient to show that L′ (V ) 6= 0 for every non-zero ‘homogeneous’ object V ∈ Cii
to show (1). In a similar way as above, we compute, for V ∈ Cii ,
(5.3)
(1i ⊠ 1i ) ⊗ L′ (V ) ⊗ (1i ⊠ 1i ) ∼
= Ai ⊗ (V ⊠ 1i ) ⊗ Ai ,
ON UNIMODULAR FINITE TENSOR CATEGORIES
23
where Ai = (1i ⊠ 1i ) ⊗ A ⊗ (1i ⊠ 1i ). The object Ai is non-zero, since
HomC env (1 ⊠ 1, Ai ) ∼
= HomC env (1i ⊠ 1i , A) ∼
= HomC (1i , 1) 6= 0.
Note that every tensorand of the right-hand side of (5.3) is an object of the tensor
full subcategory Di := Cii ⊠ Ciirev of C env . Since
EndDi (1i ⊠ 1i ) ∼
= k,
= EndC (1i ) ⊗k EndC (1i ) ∼
= k ⊗k k ∼
the unit object of Di is simple. Thus, by (2.10), the right-hand side of (5.3) is a
non-zero object whenever V 6= 0.
Corollary 5.3. Both L and R are faithful if EndC (1) ∼
= k.
5.4. The distinguished invertible object as an end. Using the equivalence Φ
given by (4.6), we also obtain the following formula of the distinguished invertible
object:
Z
Lemma 5.4. D ∼
HomC (X, 1) · X.
=
X∈C
Proof. Let iX : X ⊠ ∗X → A be the universal dinatural transformation. Since the
duality is an anti-equivalence, the family {(i∗X )∗ } is an end. Symbolically,
Z
Z
X ⊠ ∗∗∗X.
(∗ X ⊠ ∗∗X)∗ ∼
A∗ ∼
=
=
X∈C
X∈C
By Lemma 4.2, A∗ ∼
= Φ(D ⊗ (−)). Hence,
= Hom(1, D) ∼
Z
Z
D∼
HomC (∗∗ X, 1) · X ∼
HomC (X, 1) · X.
= Φ(A∗ )(1) ∼
=
=
X∈C
X∈C
The above lemma yields another proof of [17, Theorem 6.1]:
Lemma 5.5. If the unit object 1 ∈ C is a simple object, then D is isomorphic to
the socle of the projective cover of 1.
Proof. Set B(X, Y ) = HomC (X, 1) · Y . We denote by π : D −−→ B the universal
dinatural transformation of the end. For X1 , X2 ∈ C, we have
..
πX1 ⊕X2 = B(idX1 ⊕X2 , i1 p1 ) ◦ πX1 ⊕X2 + B(idX1 ⊕X2 , i2 p2 ) ◦ πX1 ⊕X2
= B(p1 , i1 ) ◦ πX1 + B(p2 , i2 ) ◦ πX2 ,
where ir : Xr → X1 ⊕ X2 and pr : X1 ⊕ X2 → Xr (r = 1, 2) are the inclusion and
the projection, respectively. Hence, if πX1 = 0 and πX2 = 0, then πX1 ⊕X2 = 0.
Now let V0 = 1, V1 , . . . , Vn be a complete set of representatives of the isomorphism classes of simple objects of C, and let Pi be the projective cover of Vi . Suppose
that πPi = 0 for all i. Then, by the above argument, πP = 0 for all projective object
P ∈ C. For each X ∈ C, there are a projective object P ∈ C and an epimorphism
f : P → X. Since B(f, X) is monic and B(f, idX ) ◦ πX = B(idP , f ) ◦ πP = 0, we
have πX = 0. Hence D = 0 by the universal property. This contradicts to the fact
that D is a simple object.
By the above argument, πPi 6= 0 for some i. Since B(Pi , Pi ) = 0 for i 6= 0,
the morphism πP0 must be non-zero. Since D is a simple object, the morphism π0
induces a monomorphism from D to the socle S of P0 . On the other hand, it is
known that S is simple [19]. Thus D ∼
= S.
24
KENICHI SHIMIZU
We consider the case where 1 ∈ C may not be a simple object. As in (2.8), we
decompose 1 ∈ C as 1 = 11 ⊕ · · · ⊕ 1m , and define Cij ⊂ C as before. Note that each
Cii is a finite tensor category. Since Cii ⊠ Ciienv is a tensor full subcategory of C env , it
is also a finite tensor category, and therefore the distinguished invertible object of
Cii is defined. We denote it by Di ∈ Cii . The following theorem and Corollary 5.8
below have been conjectured in Remark 4.3.7 of [12].
Theorem 5.6. D ∼
= D1 ⊕ · · · ⊕ Dm .
L
Proof. We decompose D as D = ij aij (aij ∈ Cij ). Since D is invertible,
M
11 ⊕ · · · ⊕ 1m ∼
aip ⊗ a∗jp .
=1∼
= D ⊗ D∗ ∼
=
i,j,p
From this, we see that there exists a permutation σ on {1, . . . , m} such that
∼
ai,σ(i) ⊗ a∗
= 1i and aij = 0 (j 6= σ(i)).
i,σ(i)
4
By the Radford S -formula (5.1), we also have
∼
∼
ai,σ(i) ∼
= D ⊗ 1σ(i) ∼
= 1∗∗∗∗
σ(i) ⊗ D = 1σ(i) ⊗ D = aσ(i),σ2 (i)
for i = 1, . . . , m. Thus σ must be the trivial permutation. In conclusion,
′
D = D1′ ⊕ · · · ⊕ Dm
for some invertible object Di′ ∈ Cii .
Now we show Di′ ∼
= Di for i = 1, . . . , m. By Lemma 5.4, we have
Z
Z
Di ∼
HomCii (X, 1i ) · X ∼
HomC (X, 1) · X.
=
=
X∈Cii
X∈Cii
By the universal property, there exists a morphism φi : D → Di in C compatible
with the coend structures. If φi = 0, then we would obtain Di = 0 by a similar
argument as in the proof of Lemma 5.5. Thus φi 6= 0. Since Di ∈ Cii , the morphism
φi induces a non-zero morphism Di′ → Di . Since Di and Di′ are invertible objects
of Cii , they are simple. Hence, by Schur’s lemma, Di ∼
= Di′ .
The following corollary is a combination of Lemma 5.5 and Theorem 5.6:
Corollary 5.7. For a finite tensor category C with property (4.1), the distinguished
invertible object of C is isomorphic to the socle of the projective cover of the unit
object of C.
We now have the following generalization of [17, Corollary 6.4]:
Corollary 5.8. A semisimple finite tensor category with property (4.1) is unimodular.
6. Applications, II. Constructions of topological invariants
6.1. General assumptions. Throughout this section, C is a finite tensor category
over a field k satisfying (4.1) and
(6.1)
EndC (1) ∼
= k.
Note that this assumption implies that 1 ∈ C is simple. Unless otherwise noted, D,
U , L, and R have the same meaning as in §5.1.
ON UNIMODULAR FINITE TENSOR CATEGORIES
25
6.2. A commutative algebra in the center. The aim of this section is to give
applications of our results to some constructions of topological invariants. As a
preparation, we consider the algebra B := R(1) obtained as the image of the trivial
algebra 1 under the monoidal functor R. For V ∈ C, we denote by R(V )B ∈ Z(C)B
the object R(V ) endowed with the right B-action given by
R2 (V,1)
R(V ) ⊗ B = R(V ) ⊗ R(1) −−−−−−−−−→ R(V ⊗ 1) = R(V ).
We note that Z(C) acts on C by
X = V = U (X) ⊗ V
(X ∈ Z(C), V ∈ C).
Theorem 6.1. The algebra B has the following properties:
(1) B is a commutative algebra in Z(C).
(2) The following functor is an equivalence of Z(C)-module categories:
K : C → Z(C)B ,
V 7→ R(V )B
(V ∈ C).
(3) (B B)∗ ∼
= R(D∗ )B as right B-modules.
(4) B is a Frobenius algebra if and only if C is unimodular.
Proof. Part (1) seems to be well-known; see, e.g., the proof of [8, Lemma 3.5]. To
prove Part (2), we note that the internal Hom functor for the Z(C)-module category
C is given by Hom(V, W ) = R(W ⊗ V ∗ ) (see Example 2.10). Corollaries 4.9 and 5.3
imply that Hom(1, −) is exact and faithful. Thus, by Theorem 2.12, the functor K
is an equivalence of Z(C)-module categories.
By Corollary 4.8, U ⊣ R ⊣ U ′ := D∗ ⊗ U (−). Let FB : Z(C)B → Z(C) be the
forgetful functor. Obviously, FB ◦ K = R. Since K is an equivalence, we have
(6.2)
K ◦ U ⊣ FB ⊣ K ◦ U ′ .
The functor (−) ⊗ (BB )∗ is also right adjoint to FB by Lemma 2.1. Hence there
exists a natural isomorphism
K(D∗ ⊗ U (X)) ∼
= X ⊗ (B B)∗
(X ∈ Z(C))
of B-modules. Part (3) is proved by letting X = 1 in the above formula. Part (4)
follows from the following logical equivalences:
(6.2)
B is Frobenius ⇐⇒ FB is Frobenius ⇐⇒ U ∼
= U ′ ⇐⇒ D ∼
= 1.
Remark 6.2. Suppose that C is unimodular. By the above theorem, there exists a
morphism λ0 : B → 1 such that (B, λ0 ) is a Frobenius algebra. It is easy to see
that λ0 6= 0 and (B, cλ0 ) is Frobenius for any c ∈ k × . Since
dimk HomZ(C) (B, 1) = dimk HomC (D, 1) = dimk EndC (1) = 1
by Corollary 4.8 and our assumption (6.1), we have the following conclusion: The
pair (B, λ) is a commutative Frobenius algebra in Z(C) for any non-zero morphism
λ : B → 1 in Z(C).
26
KENICHI SHIMIZU
∪:2→0
∩:0→2
Y:2→1
X+ : 2 → 2
X− : 2 → 2
Figure 2. Elementary handlebody-tangles
6.3. Invariants of handlebody-links. Let N0 = {0, 1, 2, . . . } denote the set of
non-negative integers. A handlebody of genus g ∈ N0 is a 3-manifold obtained
from a 3-ball by attaching g handles. For each n ∈ N0 , we fix a subset Dn ⊂ R2
consisting of n disjoint unit disks whose center lies on the x-axis. For n, m ∈ N,
an (n, m)-handlebody-tangle is a disjoint union T = T1 ⊔ · · · ⊔ Tr of handlebodies
embedded into R2 × [0, 1] such that
T ∩ (R2 × {1}) = Dn ,
T ∩ (R2 × {0}) = Dm ,
and the intersection of every genus zero component of T and R2 × {0, 1} consists
of more than two disks. A handlebody-link is a (0, 0)-handlebody-tangle. By convention, we regard the empty set as a handlebody-link.
Ishii and Masuoka [23] introduced the braided monoidal category T of handlebodytangles. By definition, the class of objects of T is the set N0 , and the set of
morphisms from n to m in T is the set of equivalence classes of (n, m)-handlebodytangles (here, two handlebody-tangles are said to be equivalent if one can be transformed into the other by a boundary-preserving isotopy R2 × [0, 1]). The composition of morphisms, the tensor product and the braiding of T are defined in a similar
way as the category of ordinary (framed) tangles.
Now let B be a braided monoidal category with braiding σ. By the definition of the category T , a braided monoidal functor T → B yields an invariant of
handlebody-links with values in EndB (1). To construct such a functor, Ishii and
Masuoka [23] introduced the following notion:
Definition 6.3. A quantum-commutative quantum-symmetric algebra (QCQSA)
in B is a triple (A, m, e) consisting of an object A ∈ B and morphisms
m : A ⊗ A → A and e : A ⊗ A → 1
satisfying the following conditions:
(Q1) m ◦ (m ⊗ idA ) = m ◦ (idA ⊗ m)
(Q2) e ◦ (m ⊗ idA ) = e ◦ (idA ⊗ m).
(Q3) m is commutative, i.e., m ◦ σA,A = m.
(Q4) e is symmetric, i.e., e ◦ σA,A = e.
(Q5) There exists a morphism c : 1 → A ⊗ A such that the triple (A, e, c) is a
left dual object of A.
It is easy to see that the monoidal category T is generated by handlebody-tangles
∪, ∩, Y, X+ and X− depicted in Figure 2. The fundamental relations among these
generators are completely determined in [22, 23]. As a result, a QCQSA (A, m, e)
in B yields a unique (up to isomorphism) braided monoidal functor F : B → T
such that F (1) = A, F (∪) = e, F (∩) = c, F (Y) = m and F (X+ ) = σA,A , where c
is the morphism in (Q5) of Definition 6.3.
ON UNIMODULAR FINITE TENSOR CATEGORIES
27
If (A, λ) is a commutative Frobenius algebra in B, then (A, mA , λ ◦ mA ) is a
QCQSA in B (moreover, every ‘unital’ QCQSA is obtained in this way). Now we
suppose that C is unimodular. Then, by Theorem 6.1, the algebra B = R(1) is
commutative and Frobenius. Given a trace λ : B → 1, we denote by
FC (−; λ) : T → Z(C)
the braided monoidal functor obtained from the QCQSA (B, mB , λ ◦ mB ) by the
above construction.
Example 6.4. Let H be a finite-dimensional Hopf algebra over k with comultiplication ∆, counit ǫ and antipode S. We use the Sweedler notation, such as
∆(h) = h(1) ⊗ h(2)
and ∆(h(1) ) ⊗ h(2) = h(1) ⊗ h(2) ⊗ h(3) = h(1) ⊗ ∆(h(2) )
for h ∈ H. A Yetter-Drinfeld module over H is a left H-module M endowed with
a left H-comodule structure m 7→ m(−1) ⊗ m(0) such that
(hm)(−1) ⊗ (hm)(0) = h(1) m(−1) S(h(3) ) ⊗ h(2) m
for all h ∈ H and m ∈ M [29]. As is well-known, the center of C := H-mod can be
identified with the category H
H YD f of finite-dimensional Yetter-Drinfeld modules
over H. Under this identification, a right adjoint R : C → H
H YD of U is given as
follows: As a vector space, R(V ) = H ⊗k V for V ∈ C. The action and the coaction
of H on R(V ) are given by
h · (a ⊗ v) = h(1) aS(h(3) ) ⊗ (h(2) · v)
and a ⊗ v 7→ a(1) ⊗ a(2) ⊗ v,
respectively, for h, a ∈ H and v ∈ V . We note that the unit η r : idZ(C) → RU and
the counit εr : U R → idC of the adjunction are given by
r
ηM
(m) = m(−1) ⊗ m(0)
and εrV (a ⊗ v) = ǫ(a)v,
respectively, for m ∈ M ∈ H
H YD f , v ∈ V ∈ C and a ∈ H. The algebra B = R(k) is
identical to the one considered in [23].
Let H and B be as in Example 6.4. A linear map λ : B → k is H-colinear if
and only if h(1) λ(h(2) ) = λ(h)1 for all h ∈ H, i.e., λ is a left integral on H. Thus,
in view of Remark 6.2, a non-zero left integral λ on H is a morphism λ : B → k in
H
H YD if and only if H is unimodular.
Now we suppose that H is unimodular. Let λ : H → k be a non-zero left integral
on H. By the above argument, we obtain a braided monoidal functor
FH (−; λ) := FH-mod (−; λ)
H
∼
from T to Z(H-mod) = H YD f . Restricting this functor to EndT (0), we obtain an
invariant of handlebody-links. However, as Ishii and Masuoka observed in [23], the
invariant obtained in this way is constantly zero unless H is cosemisimple.
To obtain a meaningful invariant from non-cosemisimple H, they proposed the
following modification of the above invariant: Every handlebody-link T can be
expressed as T = ∪ ◦ T ′ for some T ′ : 0 → 2 in T . Choose such T ′ and set
(6.3)
VH (T ; λ) := ǫ ◦ FH (T ′ ; λ),
where ǫ : H → k is the counit of H. If the condition
(6.4)
λ(S(z)) = λ(z) for all z ∈ Cent(H) (:= the center of H)
is satisfied, then VH (T ; λ) does not depend on the choice of T ′ and hence VH (−; λ)
is an invariant of handlebody-links.
28
KENICHI SHIMIZU
Although it is not yet known whether VH (−; λ) is a non-trivial invariant, it is an
interesting problem to understand their construction in the setting of finite tensor
categories. To attack this problem, we utilize the central Hopf monad Z on C. Let
µ, η, Z2 , Z0 and S have the same meaning as §3.2. To formulate the condition (6.4)
in the categorical setting, we consider the following map:
S : HomC (Z(1), 1) → HomC (Z(1), 1),
α 7→ S1 ◦ Z(α∗ ).
Note that there is an isomorphism
(6.5) HomC (Z(1), 1) → End(idC ) := Nat(idC , idC ),
α 7→ (id(−) ⊗ α) ◦ ∂1 (−),
where ∂V (X) : V ⊗ X → X ⊗ Z(V ) is a natural transformation given in §2.5. The
map S has the following meaning:
Lemma 6.5. For a natural transformation ξ : idC → idC , we define
!
ξX = (ξ∗X )∗
(V ∈ C).
Then the following diagram commutes:
(6.5)
HomC (Z(1), 1)
/ End(idC )
!
S
HomC (Z(1), 1)
(6.5)
(−)
/ End(idC ).
Proof. For all α ∈ HomC (Z(1), 1) and X ∈ C, we have
(idX ⊗ S(α)) ◦ ∂1 (X) = (idX ⊗ S1 ) ◦ (idX ⊗ Z(α∗ )) ◦ ∂1 (X)
= (idX ⊗ S1 ) ◦ ∂Z(1)∗ (X) ◦ (α∗ ⊗ idX )
= ∂1 (∗X)∗ ◦ (id∗X ⊗ α)∗
∗
(by (3.5))
∗
= ((id∗X ⊗ α) ◦ ∂1 ( X)) .
Under the identification Z(C) ≈ Z C, the functor
L : C → Z(C),
V 7→ (Z(V ), µV )
is left adjoint to U . In view of the results of §2.4, we may assume R = L! . Let B be
the commutative Frobenius algebra of Theorem 6.1 with trace λ. As a categorical
counter-part of (6.4), we introduce the following condition:
(6.6)
S(α) ◦ λ∗ = α ◦ λ∗
for all α ∈ HomC (Z(1), 1).
Theorem 6.6. Notations are as above. Given a handlebody-link T , we choose a
handlebody-tangle T ′ such that T = ∪ ◦ T ′ and then set
(6.7)
VC (T ; λ) = ε1r ◦ FC (T ′ ; λ),
where εr : U R → idC is the counit of the adjunction U ⊣ R. If (6.6) is satisfied,
then VC (T ; λ) does not depend on the choice of t′ and hence VC (−; λ) is an invariant
of handlebody-links.
Proof. By the same argument as in [23, §5], the claim reduces to that
(6.8)
(ε1r ⊗ λ) ◦ β = (λ ⊗ ε1r ) ◦ β
for all β ∈ HomZ(C) (1, B ⊗ B).
ON UNIMODULAR FINITE TENSOR CATEGORIES
29
Let η ℓ and εℓ be the unit and the counit of L ⊣ U , respectively. We may assume
that the unit η r and the counit εr of U ⊣ R are given by
r
ηM
= ∗ (εℓM ∗ ) (M ∈ Z C) and εrV = ∗ (ηVℓ ∗ ) (V ∈ C),
respectively. Then (6.8) is equivalent to that
γ ◦ (λ∗ ⊗ η1ℓ ) = γ ◦ (η1ℓ ⊗ λ∗ ) for all γ ∈ HomZ (Z(1) ⊗ Z(1), 1).
(6.9)
We note that there is a canonical isomorphism
(−)∗
Θ : HomC (Z(1), 1) −−−−−−→ HomC (1, Z(1)∗ ) −−−−−−−→ HomZ (Z(1), Z(1)∗ )
L⊣U
Explicitly, Θ(α) = SZ(1) ◦ Z(µ∗1 ) ◦ Z(α∗ ) by (2.7). Now let γ : Z(1) ⊗ Z(1) → 1 be
a morphism of Z-modules. Since
γ = evZ(1) ◦ (Θ(α) ⊗ idZ(1) )
for some α : Z(1) → 1, we compute
γ ◦ (η1ℓ ⊗ λ∗ ) = evZ(1) ◦ (Θ(α) ⊗ idZ(1) ) ◦ (η1ℓ ⊗ λ∗ )
= η1∗ ◦ SZ(1) ◦ Z(µ1∗ ) ◦ Z(α∗ ) ◦ λ∗
= S1 ◦ Z(Z(η1 )∗ ) ◦ Z(µ∗1 ) ◦ Z(α∗ ) ◦ λ∗
∗
∗
= S1 ◦ Z(α ) ◦ λ
(by the naturality of S)
(by µ1 ◦ Z(η1 ) = idZ(1) )
∗
= S(α) ◦ λ ,
∗
ℓ
γ ◦ (λ ⊗ η1 ) = λ∗∗ ◦ SZ(1) ◦ Z(µ1∗ ) ◦ Z(α∗ ) ◦ η1
= S1 ◦ Z(Z(λ∗ )∗ ) ◦ Z(µ∗1 ) ◦ Z(α∗ ) ◦ η1
∗
∗
(by the naturality of S)
∗
= S1 ◦ Z((µ1 ◦ Z(λ )) ) ◦ Z(α ) ◦ η1
= S1 ◦ Z((λ∗ ◦ Z0 )∗ ) ◦ Z(α∗ ) ◦ η1
(by the Z-linearity of λ∗ )
= S1 ◦ Z(Z0∗ ) ◦ ηZ(1)∗ ◦ λ∗∗ ◦ α∗
(by the naturality of η)
∗∗
=λ
∗
◦α
(see [5, §2.3 and §3.3])
(f ∗ = f for f ∈ EndC (1)).
= α ◦ λ∗
This means that (6.9) is equivalent to (6.6).
Example 6.7. We use the same notation as in Example 6.4. Suppose that H is
unimodular. Then, as we have seen, a non-zero right integral λ on H is a trace of
the algebra B in H
H YD f . It is easy to see that (6.7) reduces to (6.3) if C = H-mod.
To see that (6.6) reduces to (6.4), we note that there are isomorphisms
HomC (Z(1), 1) ∼
= End(idC ) ∼
= Cent(H).
If z ∈ Cent(H) corresponds to f ∈ HomC (Z(1), 1), then, by Lemma 6.5,
λ∗ ◦ f = λ(z)
and λ∗ ◦ S(f ) = λ(S −1 (z))
in EndH (k) ∼
= k. Therefore (6.6) is equivalent to (6.4) if C = H-mod.
30
KENICHI SHIMIZU
6.4. Integral of the coend Hopf algebra. Suppose that the finite tensor category C has a braiding σ. Then the object
Z X∈C
X∗ ⊗ X
FC := Z(1) =
has a structure of a Hopf algebra in C given as follows: The comultiplication ∆ and
the counit ε are induced from the comonoidal structure of the Hopf monad Z. The
multiplication and the unit are defined by
m ◦ (i1 (X) ⊗ i1 (Y )) = i1 (Y ⊗ X) ◦ (idX ∗ ⊗ σX,Y ∗ ⊗Y ) and
u = i1 (1),
respectively, for X, Y ∈ C, where iV (X) : X ∗ ⊗ V ⊗ X → Z(V ) (V, X ∈ C) is the
universal dinatural transformation. We omit the description of the antipode since
we will not use it. The Hopf algebra FC is used to construct an invariant of closed
3-manifolds; see [27] and [36].
In this subsection, we give some applications of our results to integrals for FC .
We first recall the definition of an integral in a braided Hopf algebra. Let B be a
braided monoidal category, and let H be a Hopf algebra in B. A (K-based) left
integral in H [3, Definition 3.1] is a pair (K, Λ) consisting of an object K ∈ B and
a morphism Λ : K → H in B satisfying
(6.10)
mH ◦ (idH ⊗ Λ) = εH ⊗ Λ,
where mH and εH are the multiplication and the counit of H, respectively. We
denote by Iℓ (H) the full subcategory of the category of objects over H and call it
the category of left integrals in H. The category Ir (H) of right integrals in H is
defined in a similar way.
Now suppose that B is rigid and has equalizers. Then the antipode S of H is
invertible [35, Theorem 4.1]. The category Iℓ (H) has a terminal object [3, Proposition 3.1]. We write it as (Int(H), Λℓ ) and call Int(H) the object of integrals. The
first result of this subsection is the following description of the object of integrals
of the Hopf algebra FC .
Theorem 6.8. Int(FC ) ∼
= D∗
Proof. If we identify Z(C) with the category of Z-modules, then
(6.11)
L : C → Z(C), V 7→ (Z(V ), µV )
∼ U L(1) as coalgebras. By Lemma 2.5,
is a left adjoint of U . Thus, FC =
A := U R(1) ∼
= U (∗ L(1)) ∼
= ∗ U L(1) ∼
= ∗ FC
as algebras. We note that ∗ FC is also a Hopf algebra in C. Using basic properties
of integrals for braided Hopf algebras proved in [3, 27, 35], we have the following
logical equivalences:
Int(FC ) ∼
= D∗ ⇐⇒ Int(∗ FC ) ∼
= D ⇐⇒ A has a D-valued trace.
Now set B = R(1) (so that A = U (B)) and let K : C → Z(C)B be the equivalence
of Z(C)-module categories given in Theorem 6.1. Since C is braided, there exists
e ∈ Z(C) such that U (D)
e = D∗ . By Theorem 6.1, we have
an object D
e ⊗ BB
e = K(D
e = 1) ∼
(B B)∗ ∼
= K(D∗ ) = KU (D)
=D
as right B-modules. Applying U , we get an isomorphism (A A)∗ ∼
= D∗ ⊗ AA of right
A-modules. Thus A has a D-valued trace.
ON UNIMODULAR FINITE TENSOR CATEGORIES
31
Suppose that C is unimodular. Then, by Remark 6.2, there exists a unique (up
to scalar multiple) non-zero morphism Λ : 1 → Z(1) of Z-modules. The following
theorem means that the pair (1, Λ) is a ‘two-sided’ and ‘universal’ integral.
Theorem 6.9. (1, Λ) is a terminal object of both Iℓ (FC ) and Ir (FC ).
Proof. We first show that (1, Λ) is a left integral in H. By (3.4), we have
m ◦ (i1 (X) ⊗ i1 (Y )) = i1 (Y ⊗ X) ◦ (idX ∗ ⊗ σX,Y ∗ ⊗Y )
= µ1 ◦ iZ(1) (X) ◦ (idX ∗ ⊗ i1 (Y ) ⊗ idX ) ◦ (idX ∗ ⊗ σX,Y ∗ ⊗Y )
= µ1 ◦ iZ(1) (X) ◦ (idX ∗ ⊗ σX,Z(1) ) ◦ (idX ∗ ⊗ idX ⊗ i1 (Y ))
for all X, Y ∈ C. By the Fubini theorem for coends, {idX ∗ ⊗ idX ⊗ i1 (Y )}Y ∈C is a
coend of the functor (Y1 , Y2 ) 7→ X ∗ ⊗ X ⊗ Y1∗ ⊗ Y2 . Thus,
(6.12)
m ◦ (i1 (X) ⊗ idZ(1) ) = µ1 ◦ iZ(1) (X) ◦ (idX ∗ ⊗ σX,Z(1) )
for all X ∈ C. By using this formula, we compute
m ◦ (idF ⊗ Λ) ◦ i1 (X) = µ1 ◦ iZ(1) (X) ◦ (idX ∗ ⊗ σX,Z(1) ) ◦ (idX ∗ ⊗ idX ⊗ Λ)
= µ1 ◦ iZ(1) (X) ◦ (idX ∗ ⊗ Λ ⊗ idX ) ◦ (idX ∗ ⊗ σX,1 )
= µ1 ◦ Z(Λ) ◦ i1 (X)
= Λ ◦ Z0 ◦ i1 (X).
Here, the last equality follows from the assumption that Λ : 1 → Z(1) is a morphism
of Z-modules. Recall that the counit of FC is given by ε = Z0 . Hence,
m ◦ (idFC ⊗ Λ) = Λ ◦ Z0 = Λ ◦ ε = (Λ ⊗ id1 ) ◦ (id1 ⊗ ε) = Λ ⊗ ε,
i.e., the pair (1, Λ) is a left integral in FC . To show that it is also a right integral
in H, we remark the following description of the multiplication:
m ◦ (i1 (X) ⊗ i1 (Y )) = i1 (Y ⊗ X) ◦ (idX ∗ ⊗ σX,Y ∗ ⊗Y )
= i1 (Y ⊗ X) ◦ (idX ∗ ⊗Y ∗ ⊗ σX,Y ) ◦ (idX ∗ ⊗ σX,Y ∗ ⊗ idY )
= i1 (X ⊗ Y ) ◦ (σX ∗ ,Y ∗ ⊗ idX⊗Y ) ◦ (idX ∗ ⊗ σX,Y ∗ ⊗ idY )
= i1 (X ⊗ Y ) ◦ (σX ∗ ⊗X,Y ∗ ⊗ idY ).
Here, the third equality follows from the dinaturality of i1 and the well-known
∗
formula σX,Y
= σX ∗ ,Y ∗ . In a similar way as (6.12), we obtain
(6.13)
m ◦ (idZ(1) ⊗ i1 (Y )) = µ1 ◦ iZ(1) (Y ) ◦ (σZ(1),Y ∗ ⊗ idY )
for all Y ∈ C. One can show that (1, Λ) is a right integral in a similar way as above
but by using (6.13) instead of (6.12).
Now we prove that (1, Λ) is terminal both in Iℓ (FC ). By definition, there exists
a morphism f : 1 → Int(FC ) such that Λℓ ◦ f = Λ. Obviously, f 6= 0. By Schur’s
lemma and our assumption (6.1), f is an isomorphism. Thus,
(1, Λ) ∼
= (Int(FC ), Λℓ )
in the category of objects over H, i.e., (1, Λ) is a terminal object in Iℓ (FC ). One
can show that (1, Λ) is a terminal object of Ir (FC ) in a similar way.
32
KENICHI SHIMIZU
Remark 6.10. Suppose that C is a unimodular ribbon finite tensor category. By [3,
Proposition 4.10], the morphism Λ : 1 → FC in Theorem 6.9 satisfies SF ◦ Λ = Λ,
where SF is the antipode of F := FC . Moreover, we have
(idF ⊗ m) ◦ (∆ ⊗ idF ) ◦ (Λ ⊗ Λ) = (idF ⊗ m) ◦ (idF ⊗ idF ⊗ Λ) ◦ ∆ ◦ Λ
= (id ⊗ ε ⊗ Λ) ◦ ∆ ◦ Λ
= Λ ⊗ Λ.
Hence Λ is an algebraic Kirby element in the sense of Virelizier [36, Definition 2.7].
If Λ is normalizable in his sense, then it gives rise to an invariant τC (−; Λ) of
closed 3-manifolds. The invariant τC (−; Λ) may be called the Hennings-KauffmanRadford (HKR) invariant arising from C, since the original HKR invariant [21, 25]
constructed from a finite-dimensional unimodular ribbon Hopf algebra H is the
case where C = H-mod.
References
[1] M. Aguiar and S. Mahajan. Monoidal functors, species and Hopf algebras, volume 29 of CRM
Monograph Series. American Mathematical Society, Providence, RI, 2010. With forewords by
Kenneth Brown and Stephen Chase and André Joyal.
[2] B. Bakalov and A. Kirillov, Jr. Lectures on tensor categories and modular functors, volume 21
of University Lecture Series. American Mathematical Society, Providence, RI, 2001.
[3] Y. Bespalov, T. Kerler, V. Lyubashenko, and V. Turaev. Integrals for braided Hopf algebras.
J. Pure Appl. Algebra, 148(2):113–164, 2000.
[4] A. Bruguières, S. Lack, and A. Virelizier. Hopf monads on monoidal categories. Adv. Math.,
227(2):745–800, 2011.
[5] A. Bruguières and A. Virelizier. Hopf monads. Adv. Math., 215(2):679–733, 2007.
[6] A. Bruguières and A. Virelizier. Quantum double of Hopf monads and categorical centers.
Trans. Amer. Math. Soc., 364(3):1225–1279, 2012.
[7] S. Caenepeel, G. Militaru, and S. Zhu. Frobenius and separable functors for generalized
module categories and nonlinear equations, volume 1787 of Lecture Notes in Mathematics.
Springer-Verlag, Berlin, 2002.
[8] A. Davydov, M. Müger, D. Nikshych, and V. Ostrik. The Witt group of non-degenerate
braided fusion categories. J. Reine Angew. Math., 677:135–177, 2013.
[9] B. Day. Construction of biclosed categories. PhD thesis, University of New South Wales,
1970.
[10] B. Day and R. Street. Centres of monoidal categories of functors. In Categories in algebra,
geometry and mathematical physics, volume 431 of Contemp. Math., pages 187–202. Amer.
Math. Soc., Providence, RI, 2007.
[11] P. Deligne. Catégories tannakiennes. In The Grothendieck Festschrift, Vol. II, volume 87 of
Progr. Math., pages 111–195. Birkhäuser Boston, Boston, MA, 1990.
[12] C. L. Douglas, C. Schommer-Pries, and N. Snyder. Dualizable tensor categories.
arXiv:1312.7188, 2013.
[13] C. L. Douglas, C. Schommer-Pries, and N. Snyder. The balanced tensor product of module
categories. arXiv:1406.4204, 2014.
[14] S. Eilenberg. Abstract description of some basic functors. J. Indian Math. Soc. (N.S.), 24:231–
234 (1961), 1960.
[15] S. Eilenberg and J. C. Moore. Adjoint functors and triples. Illinois J. Math., 9:381–398, 1965.
[16] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor categories. Lecture notes for MIT
18.769, 2009. http://www-math.mit.edu/~ etingof/tenscat1.pdf.
[17] P. Etingof, D. Nikshych, and V. Ostrik. An analogue of Radford’s S 4 formula for finite tensor
categories. Int. Math. Res. Not., (54):2915–2933, 2004.
[18] P. Etingof, D. Nikshych, and V. Ostrik. On fusion categories. Ann. of Math. (2), 162(2):581–
642, 2005.
[19] P. Etingof and V. Ostrik. Finite tensor categories. Mosc. Math. J., 4(3):627–654, 782–783,
2004.
ON UNIMODULAR FINITE TENSOR CATEGORIES
33
[20] I. L. Franco. Tensor products of finitely cocomplete and abelian categories. Journal of Algebra,
396(0):207 – 219, 2013.
[21] M. Hennings. Invariants of links and 3-manifolds obtained from Hopf algebras. J. London
Math. Soc. (2), 54(3):594–624, 1996.
[22] K. Ishihara and A. Ishii. An operator invariant for handlebody-knots. Fund. Math.,
217(3):233–247, 2012.
[23] A. Ishii and A. Masuoka. Handlebody-knot invariants derived from unimodular Hopf algebras.
J. Knot Theory Ramifications, 23(7):1460001, 24, 2014.
[24] C. Kassel. Quantum groups, volume 155 of Graduate Texts in Mathematics. Springer-Verlag,
New York, 1995.
[25] L. H. Kauffman and D. E. Radford. Invariants of 3-manifolds derived from finite-dimensional
Hopf algebras. J. Knot Theory Ramifications, 4(1):131–162, 1995.
[26] G. M. Kelly. Doctrinal adjunction. In Category Seminar (Proc. Sem., Sydney, 1972/1973),
pages 257–280. Lecture Notes in Math., Vol. 420. Springer, Berlin, 1974.
[27] T. Kerler and V. V. Lyubashenko. Non-semisimple topological quantum field theories for
3-manifolds with corners, volume 1765 of Lecture Notes in Mathematics. Springer-Verlag,
Berlin, 2001.
[28] S. Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in
Mathematics. Springer-Verlag, New York, second edition, 1998.
[29] S. Montgomery. Hopf algebras and their actions on rings, volume 82 of CBMS Regional
Conference Series in Mathematics. Published for the Conference Board of the Mathematical
Sciences, Washington, DC, 1993.
[30] S.-H. Ng and P. Schauenburg. Higher Frobenius-Schur indicators for pivotal categories. In
Hopf algebras and generalizations, volume 441 of Contemp. Math., pages 63–90. Amer. Math.
Soc., Providence, RI, 2007.
[31] V. Ostrik. Module categories, weak Hopf algebras and modular invariants. Transform.
Groups, 8(2):177–206, 2003.
[32] D. E. Radford. The order of the antipode of a finite dimensional Hopf algebra is finite. Amer.
J. Math., 98(2):333–355, 1976.
[33] K. Shimizu. The pivotal cover and Frobenius-Schur indicators. To appear in Journal of Algebra.
[34] K. Shimizu. The relative modular object and Frobenius extensions of finite Hopf algebras.
arXiv:1412.0211, 2014.
[35] M. Takeuchi. Finite Hopf algebras in braided tensor categories. J. Pure Appl. Algebra,
138(1):59–82, 1999.
[36] A. Virelizier. Kirby elements and quantum invariants. Proc. London Math. Soc. (3),
93(2):474–514, 2006.
[37] C. E. Watts. Intrinsic characterizations of some additive functors. Proc. Amer. Math. Soc.,
11:5–8, 1960.